National Academies Press: OpenBook

Twenty-Third Symposium on Naval Hydrodynamics (2001)

Chapter: On the Flow Structure, Tip Leakage Cavitation Inception and Associated Noise

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Suggested Citation:"On the Flow Structure, Tip Leakage Cavitation Inception and Associated Noise." National Research Council. 2001. Twenty-Third Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10189.
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Suggested Citation:"On the Flow Structure, Tip Leakage Cavitation Inception and Associated Noise." National Research Council. 2001. Twenty-Third Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10189.
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Suggested Citation:"On the Flow Structure, Tip Leakage Cavitation Inception and Associated Noise." National Research Council. 2001. Twenty-Third Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10189.
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Suggested Citation:"On the Flow Structure, Tip Leakage Cavitation Inception and Associated Noise." National Research Council. 2001. Twenty-Third Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10189.
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Suggested Citation:"On the Flow Structure, Tip Leakage Cavitation Inception and Associated Noise." National Research Council. 2001. Twenty-Third Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10189.
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Suggested Citation:"On the Flow Structure, Tip Leakage Cavitation Inception and Associated Noise." National Research Council. 2001. Twenty-Third Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10189.
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Suggested Citation:"On the Flow Structure, Tip Leakage Cavitation Inception and Associated Noise." National Research Council. 2001. Twenty-Third Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10189.
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Suggested Citation:"On the Flow Structure, Tip Leakage Cavitation Inception and Associated Noise." National Research Council. 2001. Twenty-Third Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10189.
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Suggested Citation:"On the Flow Structure, Tip Leakage Cavitation Inception and Associated Noise." National Research Council. 2001. Twenty-Third Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10189.
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Suggested Citation:"On the Flow Structure, Tip Leakage Cavitation Inception and Associated Noise." National Research Council. 2001. Twenty-Third Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10189.
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Suggested Citation:"On the Flow Structure, Tip Leakage Cavitation Inception and Associated Noise." National Research Council. 2001. Twenty-Third Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10189.
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Suggested Citation:"On the Flow Structure, Tip Leakage Cavitation Inception and Associated Noise." National Research Council. 2001. Twenty-Third Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10189.
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Suggested Citation:"On the Flow Structure, Tip Leakage Cavitation Inception and Associated Noise." National Research Council. 2001. Twenty-Third Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10189.
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Suggested Citation:"On the Flow Structure, Tip Leakage Cavitation Inception and Associated Noise." National Research Council. 2001. Twenty-Third Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10189.
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Suggested Citation:"On the Flow Structure, Tip Leakage Cavitation Inception and Associated Noise." National Research Council. 2001. Twenty-Third Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10189.
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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 ON THE FLOW STRUCTURE, TIP LEAKAGE CAVITATION INCEPTION AND ASSOCIATED NOISE 639 On the Flow Structure, Tip Leakage Cavitation Inception and Associated Noise Shridhar Gopalan1, Henry L Liu2, Joseph Katz1 (1The Johns Hopkins University, USA 2Naval Surface Warfare Center, USA) ABSTRACT The objective of this study is to investigate the effect of gap size on the flow structure and on the inception of tip leakage cavitation. Controlled cavitation tests were performed after de-aerating the water in the tunnel and using electrolysis to generate cavitation nuclei. The experiments consisted of simultaneously detecting cavitation inception a 2000fps digital camera (visual) and two accelerometers (“acoustic”) mounted on the test-section windows. Good agreement between these methods was achieved when the visual observations were performed carefully. Portions of the signal containing cavitation noise were analyzed using Hilbert and Wavelet transforms. In order to obtain the time dependent spectra, rates of cavitation events as a function of the cavitation index (σ) for the 3 gap sizes (0.6, 1.4, 2.6mm) were measured. The observations clearly demonstrate that high amplitude noise spikes are generated when the bubbles are distorted and “shredded”—broken to several bubbles following their growth in the vortex core. Mere changes to bubble size and shape caused significantly lower noise. High resolution Particle Image Velocimetry (a vector spacing of 180µm) was used to measure the flow, especially to capture the slender tip vortices where cavitation inception was observed. Seventy instantaneous realizations for the 0.6mm gap and 65 for the 1.4mm gap were analyzed to obtain distributions of circulation of the leakage vortex. PIV experiments for the 2.6mm gap are presently underway and only instantaneous samples are presented. The vortex core diameter was found to be 3–4 vector spacings. Minimum pressure coefficients in the cores of these vortices were estimated using a Rankine model. These coefficients showed a very good agreement with the measured cavitation inception indices. INTRODUCTION Cavitation occurs in liquid flows when a nucleus (bubbles, particles etc.) is captured in a region where the pressure is lower or equal to the vapor pressure (Arndt 1981, Brennen 1995, Joseph 1998). Such low-pressure regions could be at the cores of vortical structures, which occur very frequently in shear flows (Katz & O'Hern 1986, O'Hern 1991, Ran & Katz 1994, Gopalan et al. 1999, Belahadji et al. 1995). In such cases inception of cavitation is marked by intermittent events. Experimental studies on tip vortex formation and cavitation have been addressed (for e.g.) by Maines and Arndt (1997), Higuchi et al. (1989) and a numerical study of steady-state tip vortex has been reported by Hsiao & Pauley (1998). Several papers in recent years have dealt with cavitation in tip leakage or tip clearance flows. As a result of the gap, a tip leakage vortex develops which is prone to cavitation (Farrell & Billet 1994, Boulon et al. 1999). Farrell and Billet (1994) examined the effect of gap size on tip leakage cavitation and found that the cavitation inception indices increased with decreasing gap sizes. They also found a cavitation inception index minima near λ≈0.2 (λ is the ratio of the tip gap size to the maximum tip thickness). Conversely, experiments performed by Boulon et al. (1999) do not show a minimum in the cavitation inception index. Their observations could be explained using a potential flow model (elaborated in Boulon et al. and briefly in section 4 of this paper). In the present study the following issues will be addressed- (i) cavitation inception measurements using both visual and acoustic techniques, (ii) bubble dynamics during cavitation using a high-speed camera, (iii) a comparison between the acoustic signal and the occurrence of cavitation, including detailed spectral analysis of the signal, (iv) structure of the leakage flow using Particle Image Velocimetry (PIV) and the effect of gap size on leakage flow characteristics. Plots of cavitation index, σ against rate of cavitation events (rc) are obtained in nuclei controlled conditions. Three gap sizes of 0.6, 1.4 and 2.6 were studied (λ=0.12, 0.28, 0.52). Cavitation events were recorded using accelerometers attached to windows of the test section. A high-speed camera at 2000fps was used to record the motion of bubbles as they interacted with the core of the tip vortex. The observations demonstrate clearly the authoritative version for attribution.

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 ON THE FLOW STRUCTURE, TIP LEAKAGE CAVITATION INCEPTION AND ASSOCIATED NOISE 640 that high noise spikes occur when the bubbles break up in the vortex core. Oscillation in bubble size and shape cause significantly lower amplitude signals. PIV experiments were conducted using an inclined light sheet so that a cross- section of the tip leakage vortex could be captured. One purpose of these measurements is to estimate minimum pressure coefficients from the circulation and compare it to the experimental cavitation indices. The results show a very good agreement. Figure 1: (a) Experimental facility and (b) close-up of the test-section. EXPERIMENTAL SETUP AND PROCEDURE The experiments were performed in a specially designed water tunnel located at Johns Hopkins University (figure 1a). The 6.35 x 5.08 cm2 test section has a minimum length of 41 cm and maximum entrance velocity of 13 m/s. It has windows (made of optical grade lucite) on four sides to enable easy access for PIV and holographic measurements. The constant chord hydrofoil with a chord length of 50mm (and a span of 50 mm) was attached to a side window and its tip had a small clearance with the other side window (figure 1b). The maximum tip thickness was 5mm (at mid-chord) and was loaded towards the tip. Figure 2 shows the lift coefficient for this hydrofoil at 0° incidence angle. The clearance (or gap) size was varied by varying the thickness of the side window. Boundary layer suction and tripping were installed on the wall near the tip as shown in figure 1b (side view) to generate a fully developed turbulent boundary layer on the wall. The flow was driven by two 15 HP centrifugal pumps located about 4 m below the nozzle in order to prevent pump cavitation. In this study the free stream velocity in the test-section was fixed at 5 m/s (Rec, based on chord (c) equal to 2.5x105) and the cavitation index was controlled by varying the ambient pressure in the test chamber. The air content was reduced to about 3 ppm by keeping the facility under vacuum for extended periods and the dissolved oxygen content determined using an oxygen meter. The cavitation nuclei were supplied by electrolysis with two vertical wires, located upstream of the test section next to the honeycombs shown in figure 1a. The bubble generation rate (approximately 2500/ s) could be controlled by varying the current through the electrodes. The nuclei size distribution generated by this setup was measured (figure 4) and varied between 50–250µm with a median at approximately 100µm. Two accelerometers (PCB309A) with a resonant frequency of 120kHz were used to detect cavitation events (figure 3a); one was attached to the side window and the other to the bottom window, both at the vicinity of the blade trailing edge. A high-speed the authoritative version for attribution. Figure 2: Lift distribution for the hydrofoil, span=50mm.

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 ON THE FLOW STRUCTURE, TIP LEAKAGE CAVITATION INCEPTION AND ASSOCIATED NOISE 641 camera (Kodak Ektapro EM Motion Analyzer, Model 1012) at 2000fps was used to record cavitating bubbles in the tip leakage vortex. These frames were coupled with the accelerometer signals, providing a good correlation between the physical appearance of the bubbles during cavitation and the accelerometer signals. A Data Translation A-D board capable of sampling rates up to 1.33Mhz was used, and data was acquired at 250kHz/channel using LabView software. The accelerometer signals were analyzed using an in-house code, so that cavitation events could be counted (a sample in figure 5) and high-speed frames could be tagged with the noise signals (a sample in figure 7b). Figure 3: Setup for (a) visual and acoustic detection of cavitation experiments; (b) PIV experiments. PIV experiments were performed with a dual-head Nd:YAG laser rated at 30mJ/pulse (figure 3b). An inclined light sheet (shown in figure 3b) was necessary to measure the strength of the leakage vortices. When an angled sheet (at an angle α) is used in a medium (i.e. water and bounded by material of different refractive index, i.e. lucite) that is different than that of the recording device (i.e. camera in air), a proper interface at an angle, γ, given by tan γ/tan (where n is the refractive index) needs to be created. The triangular canister attached to the side window is made of Lucite and filled with Dow Corning 550 fluid, which has a refractive index of 1.5 (same as lucite), creating the interfaces shown in figure 3b. A 2K x 2K pixel2 camera with hardware-based image shifting was used to record doubly exposed (pulse separation, 51µs), densely seeded PIV images. A color filter as indicated in figure 3b was used to minimize the incident light, allowing only the emitted light from the fluorescent tracer particles to be recorded (Sridhar & Katz 1995, Gopalan & Katz 2000). An in-house developed code (Dong et al. 1992, Roth et al. 1995, Roth & Katz 2000) was used to analyze the images, initially with a 64 x 64 pixel2 interrogation window and 32 pixel spacing. Then using the output of the first run as a “guess input” for 32 x 32 pixel2 interrogation windows and 16 pixel spacing, denser velocity distributions were obtained. Such an approach is feasible only when there is enough information in a 32 x 32-pixel window, requiring dense particle seeding. With such an approach, very high-resolution velocity fields with vector spacing of 180µm could be obtained. Such measurements are essential since tip vortex core diameters are of the order of 1mm. the authoritative version for attribution. Figure 4: Cavitation nuclei size distribution measured just upstream of the leading edge of the hydrofoil.

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 ON THE FLOW STRUCTURE, TIP LEAKAGE CAVITATION INCEPTION AND ASSOCIATED NOISE 642 Figure 5: A sample accelerometer signal showing spikes caused by cavitation. CAVITATION INCEPTION INDICES AND BUBBLE DYNAMICS Accelerometer signals like the one shown in figure 5 were analyzed to obtain plots of cavitation index vs. rate of cavitation events (rc) for the three gap sizes, all at a 1° incidence angle. The results are plotted in figure 6. The cavitation index is defined as σ=(P0−Pv)/0.5ρV2 (P0 is the ambient pressure in the test section, Pv is the vapor pressure, V is the free- stream velocity, 5 m/s, and ρ is the density of water). The code used to count cavitation events (in a 10s long signal sampled at 250kHz), first identifies points greater than 1.2V then searches for amplitudes ≥3.3 V in a time interval of 0.06ms from the original point. In order to avoid counting the same event several times, the program would jump 1.4ms after finding an event and then continue. It can be seen from figure 6 that for all three cases the event rate increases with decreasing σ and both the inception indices and event rates increase with decreasing gap sizes. As an example at Figure 6: Rates of cavitation events. the authoritative version for attribution.

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 ON THE FLOW STRUCTURE, TIP LEAKAGE CAVITATION INCEPTION AND ASSOCIATED NOISE 643 10 events/s, σ for the 0.6mm gap is 11.5 as compared to 10.1 for the 1.4mm gap and 9.0 for the 2.6mm gap. The slope of the 0.6mm gap is also quite different than the 1.4 and 2.6mm gaps. Since all experiments are performed with the same nuclei distribution, the substantial differences in event rate indicates that the probability of finding low-pressure regions for the 0.6mm case is significantly higher than the 1.4mm and 2.6mm gap sizes. It is worthwhile to observe that in later stages of cavitation for the 2.6mm gap, the curve becomes much flatter (data for 1.4mm gap at these pressures were not recorded but one may expect a similar trend). This trend occurs due to the increased concentration of nuclei from prior cavitation events, a self-feeding phenomenon. Equations of power fit curves for the three gaps are also shown in figure 6. Figure 7: (a) A high-speed series (frames 1493–1498) at 2000fps (gap=0.6mm). Flow is from left to right with suction surface visible (σ~10); (b) corresponding accelerometer and strobe signals; (c) Wavelet and Hilbert transforms of the accelerometer signal. Frame timings are indicated by dashed lines. In this project we have verified that the acoustical and visual detection of cavitation match the authoritative version for attribution.

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 ON THE FLOW STRUCTURE, TIP LEAKAGE CAVITATION INCEPTION AND ASSOCIATED NOISE 644 reasonably well. In almost all cases the acoustic spikes appeared when we could detect bubbles some where in the vortex core. Three random samples of high-speed image series at time intervals of 0.5ms are presented in figures 7, 8 and 9 (top views) for the three gap sizes. The corresponding accelerometer signals are shown in figures 7b, 8b and 9b. We have carefully examined several matches between “acoustically” sensed cavitation and visually observed cavitation. The differences in bubble size and noise signals are not characteristic to their respective gap sizes, i.e. bubbles of all sizes in the range shown appeared in all gap sizes. In figure 7a, the cavitation noise begins at frame 1495, where high amplitude noise is emitted by the bubble on the left as it grows gets distorted and fragmented along the vortex core (frame 1496). The bubble on the right also experiences a similar condition in frame 1497, resulting in further noise emission. Later the bubbles are merely convected with substantial reduction in the amplitude of the noise. In figure 8b the cavitation signal occurs for a very short time and is tagged to frame 733. We do not get a large signal for the bubbles shown in frames 730– 732, although the bubbles clearly change shape and size. Conversely, frame 733 shows the bubble quite distorted and fragmented and a corresponding high amplitude noise. Figure 9 shows a high-speed Figure 8: (a) A high-speed series (frames 729–734) at 2000fps (gap=1.4mm). Flow is from left to right with suction surface visible (σ~10); (b) corresponding accelerometer and strobe signals; (c) Wavelet and Hilbert transforms of the accelerometer signal. Frame timings are indicated by dashed lines. the authoritative version for attribution.

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 ON THE FLOW STRUCTURE, TIP LEAKAGE CAVITATION INCEPTION AND ASSOCIATED NOISE 645 series for the 2.6mm gap. One can observe that the bubble moves towards the wall (consistent with the trajectory of the leakage vortex for the 2.6mm gap) while changing its shape. Cavitation starts at frame 1303 and continues on till frame 1307. The peak in noise occurs at frame 1304, where the bubble is highly distorted and fragmented again similar to previous series. Figure 9: (a) A high-speed series (frames 1299–1304) at 2000fps (gap=2.6mm). Flow is from left to right with suction surface visible (σ~10); (b) corresponding Figure 10: A 0.25s exposure showing the trajectory of accelerometer and strobe signals; (c) Wavelet and the bubbly tip leakage vortex as seen in a side view Hilbert transforms of the accelerometer signal. Frame (figure 1b), for gaps of (a) 0.6mm; (b) 1.4mm; (c) timings are indicated by dashed lines. 2.6mm. Flow is from left to right. The hydrofoil with its trailing edge and tip is visible on the left edge of the images. Thus, several such high-speed series examples indicate clearly that high amplitude noise during cavitation is primarily due to the growth, distortion and fragmentation of bubbles. Merely changes in shape or volume of the bubble generate substantially weaker noise signals. Spectra of noise signals using Wavelet and Hilbert transforms (Huang et al. 1998) are shown in figure 7c, 8c and 9c. In the spectra of figure 7c several distinct peaks at 8, 12, 17, 22, 28 and 48kHz are observed whereas, in figure 8c the energy is concentrated in a very narrow band between 22–28 kHz range, with a sharp peak at 25kHz. The multiple bubbles in figure 7a apparently Figure 11: Vertical distance of the tip leakage vortex trajectory from the trailing edge (∆y). the authoritative version for attribution.

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 ON THE FLOW STRUCTURE, TIP LEAKAGE CAVITATION INCEPTION AND ASSOCIATED NOISE 646 contribute to more complex spectrum. In figure 9c peaks at 10kHz, 22–25kHz and 30 kHz can be observed. Background noise measurements showed extremely weak peaks at the bubble resonant frequencies that disappeared when the bubble generation was stopped. There were no other external noise sources in the 10–40kHz range. Figure 12: (a) A sample instantaneous velocity field (vector spacing of 180µm); (b) Zoomed-in portion of (a), highlighting the cross section of the tip vortex. Figure 10 shows an extended exposure image of the trajectory of the bubbly tip leakage vortex as seen in a side view for the three gap sizes. The hydrofoil is visible on the left side of the images. The following points can clearly be noted: (a) The vortex trajectory becomes closer (vertical distance) to the hydrofoil as the gap size is increased (results shown in figure 11); (b) the trajectories in the 0.6 and 1.4mm gaps do not show the “bump” that is clearly evident in the case of the 2.6mm gap; (c) The cause of this bump is the interaction (merging) of two vortices, one being shed from the trailing edge and the other is the tip leakage vortex. Thus, increasing the gap to 2.6mm causes shedding and interaction between multiple structures, a phenomenon that cannot be observed distinctly in the smaller gap. Realizing again the outward trajectory of the bubbles in figure 9, this motion is most likely associated with this complex flow structure. PIV RESULTS Figure 12a shows a sample instantaneous velocity field in the inclined plane (x'y, figure 3b) with a vector spacing of approximately 180µm for the 2.6mm gap (only one velocity map is shown as an example). The zoomed-in portion of this image (figure 12b) highlights the flow in the vicinity of the tip vortex. The object on the left in this map is the hydrofoil with portions of the trailing edge and tip visible. Sample vorticity distributions derived from such velocity fields are shown in figure 13 for the three gap sizes. The cross-section of the tip leakage vortex is clearly visible at x'/c=0.12; y/c= −0.18 for the 0.6mm gap, x'/c=0.1; y/c=−0.16 for the 1.4mm gap and x'/c=0.13; y/c=−0.06 for the 2.6mm gap. Keeping in mind that the local flow is generated by an interaction of a wing tip with a turbulent boundary layer, it is not surprising that instantaneous realizations contain multiple vorticity peaks. However, unlike the tip vortex peak the others are intermittent and appear in different locations in different images. The tip vortex peak appears consistently although its exact location varies slightly (figure 15). Furthermore, clearly the tip vortex cores have substantially higher overall circulation. Also, just below the hydrofoil emerging from the gap (figure 13a and b only), we see a trail of vortical structures that are weaker than the primary leakage vortex. These secondary vortices are similar to those seen by Farrell & Billet (1994). Figure 13c for the 2.6mm gap shows a vortex core quite close to the hydrofoil as expected from figure 10c. Another high vorticity region can be also observed at x'/c=0.2; y/c=−0.12 that could very well be part of the structure from the trailing edge interacting with the tip vortex. We analyzed 70 and 65 instantaneous realizations for the 0.6mm and the authoritative version for attribution. 1.4mm gaps respectively (PIV data analysis for the 2.6mm gap is currently in progress). The regions with peak vorticity where the tip leakage vortices dissect the sheet were selected and regions with vorticity higher than 500 1/s considered to be part of the vortex core. The circulation was computed from Γ=ΣωidAi, where ω is the vorticity in an elemental area dA (=180 x 180 µm2).

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 ON THE FLOW STRUCTURE, TIP LEAKAGE CAVITATION INCEPTION AND ASSOCIATED NOISE 647 Figure 14: Distribution of circulation and corresponding minimum pressure coefficients. Figure 13: Sample instantaneous vorticity distributions for gaps of (a) 0.6mm; (b) 1.4mm and (c) 2.6mm. 4. EFFECT OF GAP SIZE ON THE STRENGTH DISTRIBUTION OF TIP VORTICES AND PRESSURE MINIMA Distributions of the measured circulation normalized by the free-stream velocity and chord length are presented in figure 14. Strength analysis for the 2.6mm gap has not been completed by the paper deadline, but will be available soon. It is evident from figure 14 that the characteristic vortex strength of the 0.6mm gap is much higher than that of the 1.4mm gap. Similar trends have been observed by Boulon et al. (1999). Figure 14 also shows the estimated pressure minima coefficients (Cpmin) computed for a Rankine vortex, Cpmin=−2/π2 (Γ/Vd)2 where d is the diameter of the vortex core. The vorticity distributions show that d mostly varies between 3–4 vector spacings (i.e. 540–717µm) and sometimes 5 spacings. No significant differences in the core sizes were seen between the two gap sizes, although this statement is greatly affected by our coarse resolution. Increased resolution would clarify this point in addition to the 2.6 mm gap data. Consequently, we show the magnitudes of Cpmin for 3 and 4 vector spacings as a function of Γ. The following points can be noted from figure 14: (1) the measured cavitation indices (figure 6) are of the same the authoritative version for attribution. magnitude as the estimated minimum pressure coefficients. (2) Clearly, the characteristic circulation of the 0.6mm case is significantly higher than that of the 1.4mm gap, explaining both the differences in σi and the trends in event rates as the pressure is reduced below the inception level. For every selected pressure below the inception level the fraction of vortices with strengths

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 ON THE FLOW STRUCTURE, TIP LEAKAGE CAVITATION INCEPTION AND ASSOCIATED NOISE 648 causing the core pressure to be below that level is much higher for the 0.6mm gap. The locations of the vortex cores in all the instantaneous realizations (points of maximum vorticity in the cross-section of the vortex) are shown in figure 15 for two gap sizes (data for the 2.6mm gap is not available yet). Figure 15: Locations of the vortex cores in the plane x'y for gaps of (a) 0.6mm (b) 1.4mm. Both show substantial meandering over ranges that are much larger than the core size. This meandering is considerably larger in the 1.4mm case than that of the 0.6mm case, where it is confined to a region with diameter of 3.7mm (in the x'y plane). Even in the latter the meandering range is 7.4% of the chordlength. In most cases the y-distance of the cores from the trailing edge of the 1.4mm gap is smaller than that of the 0.6mm gap in agreement with the results presented in figures 10 and 11. Similar trends have been observed by Boulon et al. (1999) who explain this trend using a potential flow model. A vortex near a wall has an “image” that causes an induced velocity with direction from the pressure side to the suction side. With decreasing gaps the induced velocity increases, increasing as a result the y-distance of the vortex from the hydrofoil. The higher induced velocity also increases the effective incidence angle, which would in turn increase the lift. However the presently measured 24% difference in the vortex strengths for the two gaps seems to be much larger to be purely an effect of the induced velocity. Also, the presently observed increased circulation will also cause an increase in the strength of the “image” and as a result an increase in the induced velocity. Thus, both the increasing strengths and decreasing distance from the wall affect the location of the vortex. SUMMARY Tip leakage cavitation is studied in detail with three tip gap sizes of 0.6, 1.4 and 2.6mm (λ=0.12, 0.28, 0.52). Cavitation inception indices and event rates are highest for the smallest gap size and decreases with increasing gap size. High-speed image series (at 2000fps) of cavitation in the tip vortex showed good agreement between acoustic and visual cavitation. High amplitude noise peaks were observed when bubbles were highly distorted and fragmented. Much weaker signals (by an order of magnitude) when the bubbles merely change shape or size. Cavitation noise is consistently observed in the 10–40kHz range. High resolution PIV data (180 µm distance between vectors) is used for measuring the circulation, estimate the size and location of the tip leakage vortex. The results show a core size of 3–4 vector spacings, i.e. 540µm–717µm. Minimum pressure coefficients calculated using a Rankine vortex model and the measured strengths and core diameters, lead to results that are consistent with the measured cavitation indices. Increasing the gap causes reduction (by 24%) of the tip leakage vortex strength and its vertical distance from the hydrofoil. Analysis for the 2.6mm gap is underway. Observations reveal that the flow structure for the 2.6mm gap is quite different than the 1.4 and 0.6 gaps. Meandering of the authoritative version for attribution. the vortex core is substantial in all cases. ACKNOWLEDGMENTS This project has been graciously supported by the Naval Surface Warfare Center—Carderock Division. The authors would like to thank Yury Ronzhes, Yi-Chih Chow, Brian McFadden, Dr. Ed Malkiel and Dr. Jacob Karni for their contributions.

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 ON THE FLOW STRUCTURE, TIP LEAKAGE CAVITATION INCEPTION AND ASSOCIATED NOISE 649 REFERENCES ARNDT, R.E.A. 1981 Cavitation in fluid machinery and hydraulic structures. Ann. Rev. Fluid Mech. 13, 273–328. BELAHADJI, B., FRANC, J.P. & MICHEL, J.M. 1995 Cavitation in the rotational structures of a turbulent wake. J. Fluid Mech., 287, 383–403. BOULON, O., CALLENAERE, M., FRANC, J.P., MICHEL, J.M. 1999 An experimental insight into the effect of confinement on tip vortex cavitation of an elliptical hydrofoil, J. Fluid Mech., 390. BRENNEN, C.E. 1995 Cavitation and Bubble Dynamics. Oxford University Press. DONG, R., CHU, S., KATZ, J. 1992 Quantitative Visualization of The Flow Structure Within The Volute of a Centrifugal Pump, Part A: Technique, J. Fluids Eng., 114, 390–395. FARRELL, K.J. & BILLET, M.L. 1994 A correlation of leakage vortex cavitation in axial-flow pumps, Journal of Fluids Engineering, 116. GOPALAN, S., KATZ, J., KNIO, O. 1999 The flow structure in the near field of jets and its effect on cavitation inception, Journal of Fluid Mechanics; 398, 1–43. GOPALAN, S. & KATZ, J. 2000 Flow structure and modeling issues in the closure region of attached cavitation, Physics of Fluids, 12, 895–911. HIGUCHI, H., ARNDT, R.E.A., ROGERS, M.F., 1989 Characteristics of tip vortex cavitation noise, J. Fluids Engng., 111, 495–501. HUANG, N.E. et al. 1998 The empirical mode decomposition and the Hilbert spectrum for non-linear and non-stationary time series analysis. Proc. R. Soc. London 454, 903–995. HSIAO, C.T. & PAULEY, L.L. 1998 Numerical study of the steady-state tip vortex flow over a finite-span hydrofoil. Journal of Fluids Engineering, 120. JOSEPH, D.D. 1998 Cavitation and the state of stress in a flowing liquid. J. Fluid Mech. 366, 367–378. KATZ, J. & O'HERN, T.J. 1986 Cavitation in large-scale shear flows. J. Fluids Engng 108, 373–376. MAINES, B.H. & ARNDT, R.E.A. 1997 Tip vortex formation and cavitation, Journal of Fluids Engineering, 119. O'HERN, T.J. 1990 An experimental investigation of turbulent shear flow cavitation. J. Fluid Mech. 215, 365–391. RAN, B. & KATZ, J. 1991 The response of microscopic bubbles to sudden changes in ambient pressure. J. Fluid Mech. 224, 91–115. ROTH, G., HART, D. & KATZ, J. 1995 Feasibility of using the L64720 video motion estimation processor (MEP) to increase efficiency of velocity map generation for PIV, ASME/EALA Sixth International Symposium on Laser Anemometry, Hilton Head S.C. ROTH, G., & KATZ, J. Five techniques for increasing the speed and accuracy of PIV interrogation, submitted to Meas. Sci. Technol. (June 2000). SRIDHAR, G. & KATZ, J. 1995 Lift and drag forces on microscopic bubbles entrained by a vortex. Phys. Fluids 7, 389–399. the authoritative version for attribution.

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 ON THE FLOW STRUCTURE, TIP LEAKAGE CAVITATION INCEPTION AND ASSOCIATED NOISE 650 DISCUSSION Georges L.Chahine and Chao-Tsung Hsiao DYNAFLOW, INC. We are grateful to Dr. Rood and the authors of the very interesting paper “On the Flow Structure, Tip Leakage Cavitation Inception and Associated Noise”, for requesting a discussion. The authors have conducted careful experiments to investigate cavitation inception in the gap region between a foil and a solid wall. They visualized the cavities in the vortical regions and detected the acoustic signal emitted during the dynamics. They generated bubble nuclei using electrolysis. They then modified the gap size and deduced trends in cavitation inception curves. We are particularly interested in two conclusions the authors draw: 1. “High amplitude noise during cavitation is primarily due to the growth, distortion and fragmentation of the bubbles”. 2. “The acoustical and visual detection of cavitation match reasonably well”. The first conclusion confirms our original early numerical predictions for bubble dynamics in a vortical structure [1]. Commonly used spherical bubble models predict that the bubble continuously grows during capture until it reaches the “vortex axis”, then collapses later when the vortex diffuses. To the contrary, our 3D BEM numerical simulations show that bubbles larger than some characteristic size significantly distort while being captured, form reentrant ‘jets', stretch, elongate, and get split into two or more smaller bubbles. This occurs when the pressure gradient the bubble sees is large and when various parts of its surface grow with differing velocities. At some point some parts see a pressure rise while the rest of the bubble is still growing. (See Fig. 1, 2). Once a bubble becomes centered on the vortex axis it can continue to elongate and subdivide [1] producing further noise. This leads to liquid-liquid impact on some part of the bubble that can be a significant noise source. Unfortunately, this part of the problem is difficult to compute, and is a subject of our on- going efforts for ONR. The conclusions in this paper give us further confidence in our simulations. Figure 1. Bubble shape at four instances. Initial bubble center location is at 0.2 core radius. The second conclusion could be misinterpreted if one does not underline that optical and acoustical criteria for cavitation inception lead to the same answer only when one uses sophisticated techniques such as used in the paper, involving appropriately triggered micro-photography. Since this is rarely the case in practice, the acoustic criterion gives higher values for s. In addition, the authors have used an additional factor to call inception, which is the use of the number of events per second. We believe this to be a very useful approach. Figure 2. Bubble wire-frame shapes during its capture by a vortex. Figure 3. Normalized frequency spectra for various bubble sizes in a line vortex. An additional comment to add is the great importance of the initial nuclei size on the bubble dynamics in a vortex and on the generated acoustic signals and spectra. Our study [2] shows, albeit for spherical bubbles, that it is difficult to find a simple scaling to describe all bubble sizes. Instead, there appears to exist for a given vortex flow field several families of behaviors (see Fig. 3). We would like the authors to discuss this aspect of the problem since they have the authoritative version for attribution. restricted themselves to relatively large nuclei. REFERENCES 1. G.L.Chahine, “Bubble Interactions with Vortices,” Chapter 14, Vortex Flows Ed., S.Green Kluwer Academic, 1995. 2. C-T Hsiao, G.L.Chahine, and H.L.Liu, “Scaling Effects on Bubble Dynamics in a Line Vortex Flow: Prediction of Cavitation Inception and Noise,” DYNAFLOW, INC. Technical Report 98007–1, Aug, 2000.

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 ON THE FLOW STRUCTURE, TIP LEAKAGE CAVITATION INCEPTION AND ASSOCIATED NOISE 651 AUTHOR'S REPLY We thank the discussers for reading the paper and offering several comments and suggestions. It is also nice to see that some of their numerical simulations showed similar results as observed in our experiments. In response to specific questions: First the discussers are correct to point out that the optical and acoustical detection of cavitation match well only when appropriately triggered, high-speed, high magnification photography is used. In this study we intentionally focus on bubbles that require little tension to cause cavitation inception (~100µm diameter). These initial bubble sizes are smaller than the vortex core (about 1/5th of the core diameter). We have not looked at the effect of various initial bubble sizes on bubble dynamics during cavitation. But as indicated by Chahine we agree that various bubble sizes will behave differently. For very small bubbles the critical pressure leading to cavitation inception is dependent on bubble size. Larger bubbles will be deformed, distorted and fragmented by the local pressure non-uniformities. In the present study we use bubbles that require little tension to initiate cavitation, but are still significantly smaller than the core. Once cavitation starts the bubble grows substantially leading to odd shapes, fragmentation etc. the authoritative version for attribution.

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 ON THE FLOW STRUCTURE, TIP LEAKAGE CAVITATION INCEPTION AND ASSOCIATED NOISE 652 DISCUSSION K.J.Farrell and W.A.Straka Pennsylvania State University, USA I want to thank the authors for contributing this most interesting paper to the Naval Hydrodynamics symposium. The collection of detailed PIV flow field measurements, visual observations of individual bubble dynamics, and corresponding transient acoustic measurements are comprehensive and unprecedented at this small scale. It is interesting that the higher amplitude noise peaks were associated with bubble distortion and fragmentation rather than shape or size change. In my opinion, the intended relevance of the subject investigation to the “effect of gap size on the flow structure and on the inception of tip leakage cavitation” is limited and somewhat misleading due to the omission of several important features of tip leakage flows. First, the relative motion of the tip and end-wall is omitted. In a pump, the relative motion augments the leakage flow; in a turbine the motion opposes it. The so-called scraping vortex contributes vorticity to the leakage vortex; and this contribution is certainly a function of gap size. Additionally, centrifugal forces contribute to secondary flows, which can interact with the leakage vortex at the tip. Furthermore, the contemporary practice of unloading the blade tip is not a feature of the test hydrofoil. Rather, the lower half of the span is linearly unloaded from a constant lift coefficient along the upper half-span of the hydrofoil. Given these omitted features of a tip leakage flow, the study is more relevant to the effect of confinement on tip vortex cavitation inception and associated noise. This is important distinction relative to the observation of a cavitation inception index minimum at some value of normalized clearance. The authors correctly point out that Boulon et al. (1) do not observe a cavitation minimum, while Farrell and Billet (2) and Gearhart and Ross (3) do. On the basis of total pressure loss, an optimum clearance exists when the scraping, secondary, and leakage sources of circulation at the tip have a zero net sum (4) or are identically zero [the motivation of the compressor “squealer” tip, e.g. (5)]. Conversely, the data of Figure 14 suggest higher loss with the smaller clearance. From the measured cavitation event rates plotted in Figure 6, it would appear that for very low event rates that gap size may not be a significant discriminator in cavitation performance. The observation begs the often-asked question— what is the definition of cavitation? The ITTC round-robin tests clearly identified both flow and water quality as contributors to scale effects, but certainly the threshold of inception is important as well in collapsing data. Can the authors comment on this generally and on the very low event rate behavior observed in their experiment? The statistical nature of cavitation rests significantly on the probability of an appropriately sized nucleus entering the low-pressure zone in the vortex. Accordingly, would the aurthors comment on the uniformity of the nuclei distribution and the method and location of measurement? 1. Boulon, O., Callenaere, M., Franc, J.P., and Michel, J.M. “An Experimental Insight into the Effect of Confinement on Tip Vortex Cavitation of an Elliptical Hydrofoil,” J. Fluid Mechanics, Vol. 390, 1999. 2. Farrell, K.J. and Billet, M.L. “A Correlation of Leakage Vortex Cavitation in Axial-Flow Pumps,” J. Fluids Engineering, Vol. 116. 3. Gearhart, W.S. and Ross, J.R., “Tip Leakage Effects,” Applied Research Laboratory Penn State TM 83–20, 28 February 1983. 4. Lakshminarayana, B., “Methods of Predicting the Tip Clearance Flow in Axial Flow Turbomachinery,” J. Basic Engineering, September 1970, 467– 482. 5. Wisler, D.C., “Advanced Compressor and Fan Systems,” General Electric Aircraft Engine Business Group,” presented at the VKI/Penn State Short Course on “Tip Clearance Effects in Axial Turbo-Machines,” April, 1986. AUTHOR'S REPLY We thank the discussers for reading the paper and offering several comments and suggestions. The discussers are correct to point out that several features of a tip leakage flow in a rotating turbomachine are omitted—the motion of the tip relative to the end wall, effect of centrifugal forces and unloading of the tip (ours is intentionally loaded). The differences may explain why some investigators observe a minimum in cavitation index while some don't. To answer specific questions of the discussers: First the water in the test facility is deaerated and we record accelerometer signals without generating any nuclei. Under such conditions we may observe one or two cavitation events in a ten second period, the authoritative version for attribution.

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 ON THE FLOW STRUCTURE, TIP LEAKAGE CAVITATION INCEPTION AND ASSOCIATED NOISE 653 indicating that the free stream nuclei are almost eliminated. Then we “flood” the tip region with bubbles (~100µm diameter—size distribution and details are provided in the paper) at approximately 2500 bubbles/s. The measurements were made just upstream of the leading edge and had a field of view of about 3mm. The images were analyzed using a blob analysis software. Bubbles in good focus were used for size distribution. All the traces including those that are not exactly in focus were used for computing the bubble flux. Bubble with diameters of 100µm require little tension to initiate cavitation, thus bubble size is not a critical issue in the present study. Also generation of 2500 bubbles per second makes cavitation inception less sensitive to bubble populations. However, being a very turbulent flow with large coherent structures, infrequent cavitation events (less than 2 per second) are a result of “extreme” flow conditions and not typical of the flow. With increasing events per second a trend can be correctly identified. Thus, the slope of the cavitation event rate curve is important in addition to its absolute values. the authoritative version for attribution.

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"Vive la Revolution!" was the theme of the Twenty-Third Symposium on Naval Hydrodynamics held in Val de Reuil, France, from September 17-22, 2000 as more than 140 experts in ship design, construction, and operation came together to exchange naval research developments. The forum encouraged both formal and informal discussion of presented papers, and the occasion provides an opportunity for direct communication between international peers.

This book includes sixty-three papers presented at the symposium which was organized jointly by the Office of Naval Research, the National Research Council (Naval Studies Board), and the Bassin d'Essais des Carènes. This book includes the ten topical areas discussed at the symposium: wave-induced motions and loads, hydrodynamics in ship design, propulsor hydrodynamics and hydroacoustics, CFD validation, viscous ship hydrodynamics, cavitation and bubbly flow, wave hydrodynamics, wake dynamics, shallow water hydrodynamics, and fluid dynamics in the naval context.

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