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In addition, partial cavity models cannot explain the breakup of sheet cavitation at the trailing edge into detached cavitation clouds. The process is inherently unsteady even for steady free stream conditions. Within a certain envelope of and the process is also periodic (Kjeldsen, et al, 1999). This creates a modulation of the trailing cloud cavitation that is highly erosive and very noisy (Arndt et al, 1997). Although considerable research has gone into avoiding cavitation, little effort has been made to understand the complex physics associated with operation in the partially cavitating regime. The occurrence of sheet cavitation and the transition to cloud cavitation results in a highly unstable flow that can induce significant fluctuations in lift, thrust and torque. In order to gain a better understanding of the complex physics involved, an integrated numerical/experimental approach was used. A 2D NACA 0015 hydrofoil was selected for study, because of its previous use by several investigators around the world. The simulation methodology is based on large eddy simulation (LES), using a barotropic phase model to couple the continuity and momentum equations. These simulations were carried out in an interactive way with the experimental efforts. For example, early simulations indicated that pressure side cavitation can occur under special conditions. This was experimentally confirmed and has led to an exhaustive investigation of cavitation induced lift oscillations. The LES approach was adopted to resolve space wise variation of the flow due to large scale eddies. The weakly compressible flow approach proposed by Song and Yuan (1988) is used to resolve the rapidly changing flows in the liquid phase and, at the same time, to speed up the computation as compared with a typical incompressible flow approach. To overcome the difficulty presented by the unsteady and multiple free surfaces, Kubota et al (1992) proposed a “bubble two- phase flow model” based on an assumption that the fluid consists of a bubble-water mixture and the bubble volume change is governed by a modified Rayleigh-Plesset equation. More recently Song et al (1997) assumed the fluid to be barotropic and the density is a continuos function of pressure in the entire liquid-vapor phase. The pioneering works of Helmholtz (1868) and Kirchhoff (1869) assumed the cavitating flow to be a steady state free-surface potential flow of the ideal fluid. They completely ignored the dynamics of the gas flow and neglected the viscosity and com-pressibility of the liquid phase. An obvious penalty of neglecting viscosity is the inability to simulate flows having strong velocity gradient. Less known fact is that the compressibility effect is important if pressure changes rapidly with time, even when Mach number is very small (see Song, 1996). By ignoring gas dy-namics, the numerical treatment of free surface boun-dary may become untenable for highly unsteady and multiple cavity cases. Unsteady cavity flows about hydrofoils are time wise and space wise highly variable two-phase flows, making it necessary to account for compressibility, viscosity as well as the gas dynamics. The experiments were carried at two different scales in two different water tunnels, at the Saint Anthony Falls Laboratory (SAFL) and at the Versuchsanstalt für Wasserbau (VAO) in Obernach, Germany. The tests were designed to complement each other and to capitalize on the special features of each facility. THEORETICAL CONSIDERATIONS Compressibility boundary layer theory The importance of compressibility in small Mach number flows has long been overlooked. It is generally known that the compressibility effect is negligible if the flow is steady and Mach number is small, M 0.2. However, Song (1996) has shown that the compressibility can't be ignored if the flow is highly unsteady even when M is very small. The equation of state of water in liquid phase can be accurately represented by the following equation for a very wide range of pressure change. (1) In the above equation “a” is the sound speed and the subscript “o” represents a reference condition. By eliminating from the equation of continuity using Eq. 1 the following equation is obtained. (2) 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 INSTABILITY OF PARTIAL CAVITATION: A NUMERICAL/EXPERIMENTAL APPROACH 602 The equations of continuity and motion may be written in a conservative form as follows. (8) where S is a source, which may or may not exist and i, j, k are the unit vectors in x, y, and z directions, respectively. The functions U, E, F, G and S are defined in the following. (9) (10) (11) (12) (13) The source term S is identically equal to zero for the liquid phase. Numerical Approach The finite volume approach with MacCormack's (1969) predictor-corrector method has been used with good results. Eq. 2 is first averaged over a computational finite volume. In the process, the volume integral of divergence is converted to the surface integral by applying the divergence theory. The resulting equation is (14) In the above equation the superscript bar means volume averaged quantity, A is the surface area, and n is the unit normal vector on the surface of integration. Since the quantities to be calculated are always the volume averaged values, the superscript bar will be dropped hereafter. The above equation is a simple statement of the law of conservation that the rate of increase of U in a volume is equal to the net influx of plus the source S. Thus, Eq. 14 can be discretized with any standard finite difference scheme. Yuan (1986) describes detailed description of the process used herein. Since a finite volume approach can resolve only the quantity of scale larger than the finite volume used, it is necessary to use a closure model for turbulent flow simulation. A simple Smagorinsky (1963) subgrid scale turbulence model is used. This model assumes that the shear stress tensor ij can be calculated by adding an eddy viscosity t to the kinematic viscosity. Smagorinsky proposed the following equation for the eddy viscosity. (15) In the above equation C is a coefficient to be determined and is size of the finite volume. Experience indicates that a C distribution, which is 0 on the wall and increases to 0.12 outside of the viscous boundary layer generally gives satisfactory results. The size of finite volume used is often too large to adequately resolve the sharp velocity gradient near a solid wall where the viscous boundary layer is thin. For this reason a wall function or a partial-slip condition (becomes no-slip when boundary layer is thick) is used: see He and Song (1991) for a description of this condition. Because cavity is assumed to be a continuous extension of the liquid, no free surface boundary condition is necessary. Only the hydrodynamic quantities are of interest (acoustics are ignored) and, therefore, an arbitrarily large value of M=0.4 is chosen for the liquid part of the flow to speed up the computation. Recall that the outer solution of the weakly compressible flow equations is independent of M and represent the incompressible flow. Although much larger M can be used, the gain in computational speed diminishes as M is further increased. EXPERIMENTAL METHOD In order to gain a better understanding of the complex physics involved, a simple geometry was studied. A 2D NACA 0015 hydrofoil was selected because of its previous use by several investigators around the world. The experiments were carried out at 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 INSTABILITY OF PARTIAL CAVITATION: A NUMERICAL/EXPERIMENTAL APPROACH 608 Figure 12 Simulation snapshot for /2=4.1 From picture (e) the maximum sheet cavity length is estimated to be about 1.3 c. This long sheet cavity is about to be broken off by the reentrant jet as shown by picture (e). Pictures (f, a, b) show the convection of the large eddy or the cloud cavity after it is broken off. The cloud cavity collapses further downstream, which is outside of the region shown in the figure. Fig. 15 is an illustration of the pressure field at the same time increments as for the vorticity. This figure also shows evidence of shock wave produced by the collapsing cloud cavity traveling upstream. As observed experimentally, the shock wave can destroy the sheet cavity near the nose for short period of time as shown by picture (f) of Figure 14 (very thin boundary layer indicates no cavitation.) This was a surprise when first observed experimentally. In addition, Figure 13 shows that the lift oscillations also contain two peaks. In this case the primary peak is that of the sheet/cloud cavity oscillation, which is dependent on the chord length rather than the cavity length. Also in agreement with experimental observations. Wake Characteristics Cavitation has a profound effect on the wake characteristics behind a hydrofoil. During the current study, LDV measurements were made in the wake of the foil in cavitating and non-cavitating conditions. Velocity profiles were obtained a several cross-sections downstream of the trailing edge of the foil. Mean velocity data were obtained and analyzed for a variety of conditions ( and ). To date, two different types of analysis have been performed. These analyses provide not only insight into understanding the periodic phenomena cited above but also provide insight into the problem of predicting bubbly wakes due to cavitation. Figure 13. Computed variation in spectral characteristics with increasing. /2 varies from 1.75 to 5.73 Figure 14. Computed vorticity field for /2=2.15 the authoritative version for attribution. Figure 15. Numerical simulation of the pressure field during extensive cavitation, /2=2.15

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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 INSTABILITY OF PARTIAL CAVITATION: A NUMERICAL/EXPERIMENTAL APPROACH 609 The basic idea for both methods is an observed increase in measured data rate when bubbles are passing through the LDV measurement volume. This increase is, in general, greater than 10 times that for non-cavitating conditions. The wake flow behind the hydrofoil was also investigated for both cavitating and non-cavitating conditions. Using LDV measurements, it is found that self-similarity of the wake shape exists quite close to the trailing edge (x/ c=0.12), for both cavitating and non-cavitating flow. Mean velocity profiles for cavitating and non-cavitating conditions are presented in Figure 16. The data are presented in a form suggested by the similitude of turbulent wakes: (17) where x is distance from the trailing edge, y is the distance normal to the flow measured from the center of the wake c is the chord length and Uref is the velocity at the edge of the wake. It is clearly evident that the rate of spreading of the wake is significantly larger under cavitating flow conditions. This effect has been also reported by other workers in the field (Kubota et al, 1989). When cloud cavitation is present, large vortical structures containing numerous bubbles are observed to be shed into the wake. These clouds of bubbles extend much further in the cross stream direction than the viscous wake associated with non-cavitating flow. Much of the important physics in the process are obscured by the averaging process. While the non-cavitating LDV signal resembles a typical turbulence signature, the cavitating signal is skewed towards lower velocity. The strong negative fluctuations in velocity are due to the imprint left by the periodic passage of vortices from the cavitation process that extends much further from the wake centerline than a typical viscous wake. This is graphically illustrated in the numerical simulations shown in Figure 11. In fact there is excellent agreement between animated simulations and high speed videos of the vortical clouds of bubbles. The numerical simulations shown in Figure 11 are for a case close to the maximum amplitude of lift oscillations. Note the strong upwelling close to mid-span that results in strong oscillations in lift and surface pressure. It should also be pointed out that averaged numerical data also agree very well with the experimental data shown in Figure 16 for both cavitating and non-cavitating flow. In the case of cavitating flow, it is important to note that the numerical simulations fit the data close to the trailing edge, but deviate from the measurements further downstream. It is conjectured that this is an effect of dissolved gas that has come out of solution. The current numerical model considers only vapor that will condense downstream, while gas bubbles will persist in the wake. the authoritative version for attribution. Figure 16 Experimental (above) and numerical (below) wake data with/without cavitation compared. y=0 is taken to be that of maximum velocity deficit for the non-cavitating case. It is also natural to conclude that the gas

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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 INSTABILITY OF PARTIAL CAVITATION: A NUMERICAL/EXPERIMENTAL APPROACH 611 As would be expected, the spectral characteristics of the surface pressure in the region of maximum intensity should be similar the to the spectral characteristics of the surface pressure. This is shown in Figure 19 that is similar to Figure 4. Note that a similar bifurcation in the peak frequency occurs at about /2 =4. In analogy with the lift data displayed in Figure 7, spectra at two values of /2 are shown for x/c=0.74 are shown in Figure 20. Two dominant peaks are evident, with the higher frequency peak being dominate at the higher value of /2. When comparing these spectra with JTFA analyses of the same data, it is also noted that either one frequency or the other occurs, but not simultaneously. The duration of each frequency is proportional to the amplitude of that frequency component in the conventional spectrum. With lift oscillation and pressure data as a guide, the spectral characteristics of the noise can be analyzed The spectra are normalized in the following manner: (17) where is the mean square acoustic pressure and f is the frequency bandwidth. Noting that the spectral characteristics of the lift oscillations vary considerably with /2, it is interesting to compare how the spectral characteristics of the noise vary over the same range. This is shown in Figure 21 where spectra at different values of /2 are compared over the same range shown in Figure 4 for lift oscillations. It is very clear that the noise maximizes in the same region as the lift oscillations are a maximum. The noise also persists to much higher frequencies. COMPARISON WITH NON-CAVITATING SEPARATED FLOWS Numerous experiments and numerical studies of flows over a foil in a non-cavitating condition have been carried out for several decades. The general flow patterns at different attack angles are well described based on both experiments and com-putations. For the purpose of comparison and for better understanding of the mechanics of cavity flows, this numerical model has also been used to simulate non-cavitating flows about NACA0015 foil. A non-cavitating condition is simulated by assigning a very large value to the cavitation number while the Figure 19. Surface pressure spectra at l/c=0.74 Figure 18. RMS surface pressure. The intensity of the color denotes the amplitude normalized to dynamic pressure. 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 INSTABILITY OF PARTIAL CAVITATION: A NUMERICAL/EXPERIMENTAL APPROACH 612 Reynolds number is kept constant at 105. The boundary layer is fully attached when the angle of attack is small. Boundary layer separation occurs at about 90% chord when =8. The separation point creeps upstream as is increased to 11 while the separated flow remains stable. The only oscillatory motion is that of the wake instability. Figure 20. Spectral characteristics of surface pressure are compared at l/c=0.74 for two values of /2 Figure 21 Spectral characteristics of radiated sound. A slight increase of the angle of attack to 12 results in the separation point moving upstream to about 25% chord with vortex shedding taking place as shown in Fig. 22. The elongated separation bubble “a” detaches to produce a large clockwise rotating eddy “b” which convects downstream to become “c”. When eddy “b” detaches from “a” it induces a counter rotating vortex “d”. It is interesting to compare Fig. 22 with Fig. 11 and find that there is much similarity between a cavitating flow and a non-cavitating flow at a somewhat larger attack angle. In both cases, the flow 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 INSTABILITY OF PARTIAL CAVITATION: A NUMERICAL/EXPERIMENTAL APPROACH 613 is dominated by a periodic growth and detachment of a boundary layer separation bubble. The downstream part of the elongated separation bubble is a large coherent eddy dominated by negative vorticity. But there is a thin layer of positive vorticity near the foil surface shown in red. This reverse flow with positive vorticity is a reentrant jet that is responsible for the breaking away of the sheet cavity. The way that the reentrant jet terns upward and wraps around the detached large eddy can be observed in both cases. Figure 22. Simulation of non-cavitating flow at =12 SUMMARY AND CONCLUSIONS An extensive investigation of sheet/cloud cavitation was carried out using both numerical and experimental techniques. Properly normalized, the data agree in both amplitude and spectral characteristics. Data for lift, surface pressure and noise are remarkably consistent. Three flow regimes are noted roughly demarcated by the parameter /2. A strong bifurcation in the data exists at /2 where the maximum amplitude in lift, noise and pressure is noted. Excellent agreement with numerical sim-ulations was achieved. This leads to a very optimistic view of the simulation model used. Steps have been taken to significantly reduce computation time. The authors feel that this combined numerical/experimental approach shows great promise for further research. It is important to underscore the fact that /2 is a parameter whose genesis is found in linearized inviscid theory for a thin flat plate. It was a surprise that this parameter was even an approximate description of cavitating flow. Detailed examination does show that the vertical extent of the cavity flow is less at lower angle of attack even though l/c and /2 are constant. The acoustic radiation from different cavitation patterns on a NACA 0015 hydrofoil were studied. In addition, the noise characteristics were correlated with measurements of unsteady pressure on the surface of the foil. These measurements were made using a new method utilizing piezoelectric film. The noise characteristics are very similar to the characteristics of the lift oscillations. Numerical and experimental work have identified three regions of cavitating flow. Cavitation noise spectra were also found to vary significantly with the type of cavitation, e.g. bubble/patch, sheet/cloud or fully developed super-cavitation. The best correlating parameter is /2. This is parameter is only truly valid for a thin flat plate (Acosta, 1955). However even for the relatively thick foil studied here, the overall features of the flow are determined by this parameter. The acoustic pressure peaks at about /2.= 3.5. The spectra persist to high values of frequency in the range 2.5 /2 5.0. Reasonably good collapse of the data was found by normalizing in the form The surface pressure technique shows good promise for detecting regions of intense cavitation activity. The observed pressure characteristics are in good qualitative agreement with the numerical simulations and the measured lift oscillations. The array of techniques developed to study this complex flow show good promise for surface pressure/ acoustic radiation and lift/acoustic radiation correlations. ACKNOWLEDGMENTS This work was sponsored by the National Science Foundation and the Office of Naval Research. Mr Michael Levy provided invaluable assistance in obtaining the noise and pressure data. Mr. Fayi Zhou and Dr. G.Wang have also provided some help at the early stage of the program. The computer time has been provided by the Minnesota Supercomputer Institute, University of Minnesota through its grant program. REFERENCES Acosta, A.J. (1955) “A note on partial cavitation of flat plate hydrofoils” Calif. Inst. of Tech. Hydro Lab, Rep. E-19.9 Arndt, R.E.A. (1981), “Cavitation in fluid machinery and hydraulic structures”, Ann. Rev. Fluid Mech., Vol. 13. 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 INSTABILITY OF PARTIAL CAVITATION: A NUMERICAL/EXPERIMENTAL APPROACH 614 Arndt, R.E.A., Voigt, R.L. Jr., Sinclair, J., Rodrigue, P., and Ferreira, A,. (1989) “Cavitation Erosion in Hydroturbines,”, Journal of Hydraulic Engineering, ASCE, October Arndt, R.E.A., Paul, S andEllis, C.R. (1997) “Application of piezoelectric film in cavitation research” Journal of Hydraulic Engineering, Vol. 123, no. 6, June Doligalski, T.L., Smith, C.R., and Walker, J.D.A., 1994, “Vortex interactions with walls,” Ann. Rev. Fluid Mech., vol. 26, pp. 573–616. Helmholtz, H. (1868), “On Discontinuous Movements of Fluid” Phil. Mag., 36, No. 4, 337–346. Izumida, Y., Tamiya, S., and Kato, H., 1980, “The relationship between characteristics of partial cavitation and flow separation,” Proc. 10th IAHR Symp., Tokyo, pp. 168–181. Kjeldsen, M., Arndt, R.E.A. and Effertz, M., (1999) “Investigation of Unsteady Cavitation Phenomena” Proceedings of the 3rd ASME/JSME Fluids Engineering Conference, San Francisco, CA, July Kirchhoff, G. (1869), “Zur Theorie Freier Flussigkeitsstrahlen”, Jr. reine u. angew, Math., 70, 289–298. Kiya, M.,, and Sasaki, K., 1985, “Structure of large-scale vortices and unsteady reverse flow in the reattaching zone of a turbulent separation bubble,” Journal of Fluid Mechanics, vol. 154, pp. 463–491. Kubota, A., Kato, H., Yamaguchi, H., and Maeda, M., “Unsteady structure measurement of cloud cavitation on a foil section using conditional sampling technique” J. Fluids Eng., 111, 204–210 MacCormack, R.W., (1969), “The effect of Viscosity in Hypervelocity Impact Cratering”, AIAA Paper 69–354, Cincinnati, Ohio Riabouchinsky, D. (1919), “On Steady Fluid Motion with Free Surface”, Proc. London Math. Soc. 19, 206–215. Song, C.S. (1965) ‘“Two-dimensional Supercavitating Plate Oscillating Under a Free Surface”, Journal of Ship Research, Vol. 9, No. 1, 40–55. Song, C.C.S. and Yuan, M. (1988), “A weakly Compressible Flow Model and Rapid Convergence Methods”, Journal of Fluids Engineering, Vol. 110, 441–445. Song, C.C.S. (1996), “Compressibility boundary Layer Theory and its Significance in Computational Hydrodynamics”, Journal of Hydrodynamics, Series B, Vol. 8, No. 2, 92–101 Song, C.C.S. and Chen, X. (1996), “Compressibility Boundary Layer and Computation of Small Mach Number Flows”, Hydrodynamics, Theory and Applications, Proc. 2nd International Conf. on Hydrodynamics, Hong Kong, 815–820. Song, C.C.S., He, J., Zhou, F. and Wang, G. (1997), “Numerical Simulation of Cavitating and Non-cavitating Flows over a Hydrofoil,” St. Anthony Falls Laboratory, Project Report No. 402, University of Minnesota Tulin, M.P. (1958), “New Development in the Theory of Supercavitating Flows”, Proc. Second Symp. on Naval Hydrodynamics, ONR/ACR-38, pp. 235–260. Watanabe, S., Tsujimoto, Y., Franc, J.P., and Michel, J.M., (1998) “Linear Analyses of Cavitation Instabilities” Proceedings of Third International Symposium on Cavitation, Grenoble, France, April Wu, T.Y. (1956), “A Free Streamline Theory for Two-dimensional Fully Cavitated Hydrofoils”, Journal of Mathematics and Physics, 35, 236–265. Ye, Z., Gopalan, S. and Katz, J. (1998) “On the Flow Structure and Vorticity Production due to Sheet Cavitation” Proceedings of 1998 ASME Fluids Engineering Division Summer Meeting, Paper No. FEDSAM98–5301 Zaman, K.B.M.Q., McKinzie, D.J., and Rumsey, C. L., 1989, “A natural low-frequency oscillation of the flow over an airfoil near stalling conditions,” Journal of Fluid Mechanics, vol. 202, pp. 403–442. 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 INSTABILITY OF PARTIAL CAVITATION: A NUMERICAL/EXPERIMENTAL APPROACH 615 DISCUSSION L.Briancon-Marjollet Bassin d'Essais des Carenes I am not sure to well understand what you consider a bubble patch cavitation. In my opinion, you use a density/ pressure curve in your model everywhere in the fluid. So, you did not take into account statistical point of view for bubble cavitation. So, what is the accuracy of the lift time dependence curve (fig. 10) for bubble patch cavitation? AUTHOR'S REPLY Bubble cavity in this paper refers to a large detached cavity, which is generated near the nose of the foil and convected away in periodic manner. It usually occurs when the attack angle is small and the cavitation number is only slightly below the critical value. Its occurrence is quite predictable and deterministic. So the phenomenon described here is quite different from the cavitation gas nuclei which occurs randomly. Because the size of the bubble cavity is quite large, it causes the lift to oscillate at the Strouhal number based on the cord length of about 0.1. The computed lift oscillation agrees very well with the experimental data. the authoritative version for attribution.