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Experimental investigation of cavitating structures in the near wake of a cylinder
International Journal of Multiphase Flow 89 (2017) 207–217
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Experimental investigation of cavitating structures in the near wake of a cylinder Pankaj Kumar, Dhiman Chatterjee∗, Shamit Bakshi Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai 600036, India
a r t i c l e
i n f o
Article history: Received 19 February 2016 Revised 12 August 2016 Accepted 24 September 2016 Available online 11 November 2016 Keywords: Cavitation High speed images Shear layer Formation length
a b s t r a c t An experimental investigation of cavitating structures in the near-wake region of a cylinder is presented. From high-speed imaging of this subcritical flow (Reynolds number of 64,0 0 0), it is found that inception of cavities occurs in the shear layer. At the developed cavitation condition, the cavities in the separated zone and the free shear layer merge. A distinct spanwise variation in cavitation activity is observed. The non-dimensionalized correlation length at inception varies from close to a non-cavitating value of about 3.5 to about 1 at developed cavitation. The non-dimensionalized length of formation, characterized by crossover of the free shear layer and the wake axis, increases from 1 to 1.8 as the cavitation number is reduced from 85% to 50% of the inception value. A frequency analysis of the cavity dynamics indicates that although the vortex shedding frequency is dominant in the shear layer, there are peaks corresponding to other frequencies in other flow regions. The presence of a sharp peak at 125 Hz, corresponding to a Strouhal number of 0.2, along with a range of frequencies, is also verified independently through measurement of fluctuating pressure at the cylinder surface. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction Non-cavitating flow past a circular cylinder has been the subject of extensive research, since this simple geometry exhibits interesting fluid phenomena and also has wide engineering applications. This flow is classified as subcritical, critical or supercritical, depending on the value of the Reynolds number Re (Roshko, 1961; Schewe, 1983; Zdravkovich, 1997), with each flow regime exhibiting distinctive features, such as the location of the transition to turbulence (in the wake, shear layer or boundary layer) and different characteristic frequencies of pressure and velocity fluctuations. A review of the literature reveals that most investigations have been carried out at low Re and have been directed at understanding two-dimensional vortex shedding as well as the origin and growth of three-dimensional disturbances, which can be important even for nominally two-dimensional cylinders (Roshko, 1993; Williamson, 1996). The transition in the wake takes place at much lower Re. Distinct irregularities in velocity fluctuations becomes visible as Re exceeds 300 (Roshko, 1954), which Roshko (1954) characterized as the irregular regime. Bloor (1964) observed the first transition eddies at Re = 1300 and found the frequency of the transition eddies to scale as Re0.5 . Wei and Smith (1986) noted
∗
Corresponding author. Fax: +9144 22574652. E-mail address: [email protected] (D. Chatterjee).
http://dx.doi.org/10.1016/j.ijmultiphaseflow.2016.09.025 0301-9322/© 2016 Elsevier Ltd. All rights reserved.
that the formation of ‘secondary vortices’ was linked with these high-frequency fluctuations and their results showed an Re0.77 dependence. As transition proceeds to the shear layer, the coefficient of base suction pressure (−C pb ) increases, the Reynolds stress level increases and the Strouhal number gradually decreases (Norberg, 1994). With further increase in Re in the upper subcritical regime (2 × 104 < Re < 2 × 105 ), the flow features change. There have been fewer studies of this regime, which is briefly described in the next paragraph. One important characteristic of the wake is the eddy-formation region. This region has been defined differently by different researchers: as the region behind the cylinder up to the minimum of the measured Cp distribution along the wake axis (Linke, 1931), as the region where low-frequency oscillations disappear (Bloor, 1964), as the location at which the free shear layer crosses the wake axis (Bloor and Gerrard, 1966) and as the location of maximum fluctuations umax in the wake (Griffin and Votaw, 1972). The formation length (Lf ) is the distance, along the flow direction, of the eddy-formation region from the cylinder centre. This formation length (Lf ), when non-dimensionalized with the cylinder diameter (d), decreases with increasing Re in the range Re = 103 –2 × 105 and is typically less than 2 above Re = 10 0 0 0. The aspect ratio L/D of the cylinder, besides the Re of the flow, also influences this value. This shortening of Lf , coupled with a widening of the shear layer at higher Re, was first observed by Gerrard (1966). Unal and Rockwell (1988) used a flow visualization technique to reveal the
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decrease in Lf . Further increase in Re does not lead to any significant shortening of Lf . The overall organization of the flow regimes behind a cylinder in the subcritical regime was nicely shown in an experiment by Maekawa and Mizuno (1967), who used hot-wire signals to detect the free shear layer and the near-wake characteristics at Re = 65 × 103 . It was found that the signals detected in the near wake are random low-frequency velocity fluctuations, while those from the free shear layer exhibit the vortex-shedding frequency. In the region downstream of the formation region along the axis, there are two components to the signal, which signifies alternating crossing of flows from the two sides of the wake. This feature is probed further in the present work in the context of cavitating flow. The flow is inherently and highly three-dimensional, but, as indicated by Szepessy and Bearman (1992), for very small L/D values, nominally ‘two-dimensional’ conditions can be assumed. The correlation length, i.e. the length within which the pressure signals are highly correlated (with correlation coefficient ≥ 0.5) along the spanwise direction, was found to be 2.5 times the cylinder diameter for Re ࣃ 71 × 103 (Szepessy and Bearman, 1992) and 3.5 times the cylinder diameter for Re ࣃ 43 × 103 (Szepessy, 1994). In the present work, the L/D of the cylinder is similar to that in Szepessy and Bearman (1992), which helps to reveal the influence of the extent of cavitation activity on the correlation length. In comparison with non-cavitating flow, there has been relatively less research on cavitating flow past a circular cylinder. Furthermore, most investigations to date have examined the critical and supercritical flow regimes. A survey of the literature on cavitating flow past a circular cylinder shows that the major thrust of research has been on determining variations in overall cavity geometry (length and width) and drag coefficient as functions of cavitation number. For example, in one of the earliest pieces of work in this area, Shalnev (1965) obtained the cavity length and width and the drag coefficient as functions of the blockage factor in a constrained test section. The significance of the blockage factor was also discussed by Ramamurthy and Bhaskaran (1977) and Rao and Chandrasekhara (1976). The effects of cavitation number on cavity length and width as well as on vortex-shedding frequency were studied by Varga and Sebestyen (1966) for supercritical flow. Rao and Chandrasekhara (1976), on the basis of highspeed movies, noted that there were considerable longitudinal oscillations downstream of the cavity. This finding highlights the importance of characterizing local cavity oscillations in order to understand the interaction of vortices with the cavity. In a related study of local dynamics, Fry (1984) determined the cavitation noise spectrum at different downstream locations in the wake for different cavitation numbers. His results indicate that the peak noise was observed when the cavity length was twice the cylinder diameter. Matsudaira et al. (1992) measured the dynamic pressure due to bubble collapse at several locations in the wake of a cylinder for an Re range of 4.5 × 105 –6 × 105 . In the supercritical flow regime, they found that the frequency of peaks due to impulsive pressure was high at x/d ࣃ 1.3 in the separation region, while in the vortex-formation region (2.1 < x/d < 2.9), peaks were fewer and were mixed with background turbulent fluctuations. They further found that, with a decrease in cavitation number, the cavity length determined from pressure contour maps increased and extended downstream. Investigation of the local cavity dynamics using high-speed imaging forms an important part of the present work. It is clear from the literature cited above that there has been a few investigations of cavitating flow past a cylinder in the upper subcritical flow region (20, 0 0 0 < Re < 20 0, 0 0 0). Studies of noncavitating flow in this regime have led to interesting observations such as the organization of flow regimes (Maekawa and Mizuno, 1967), the phase drift and the associated three-dimensionality be-
Fig. 1. Schematic plan view of the test section with instrument. The inset shows the front view of the cylinder inside the test section. Also shown is the (x, y, z) axis system used in this paper. The origin is at the centre of the cylinder and is aligned with the lower edge of the test-section window.
yond a certain spanwise distance (Szepessy, 1994; Szepessy and Bearman, 1992). Hence the pertinent question that needs to be addressed is the way in which these observations may be modified as we move gradually from non-cavitating through weakly cavitating to a highly cavitating regime at the same value of Re. Furthermore, most work, even in the critical or supercritical regime, has been related to measurements of cylinder surface pressure and drag, overall cavity structure, and cavitation-induced noise, and has not dealt with cavity dynamics. Following Fry (1984), we suggest that although cavity length and cavity-shedding frequency have already been measured behind cavitating circular cylinders, these cannot explain the location of the peak noise value. Hence, in this work, we attempt to gain further insight by presenting results that relate the dynamics of cavitating structures and the frequency of fluctuation of cavities to the cavitation number. In Section 2, the experimental facility is described. This is followed by a description of the image processing in Section 3 and the results are then discussed in Section 4. Our conclusions are given in Section 5. 2. Experimental facility and methodology The cavitation test rig used in the present study is described in detail by Kumar et al. (2014) and in this section we describe only the major aspects. The test section is 65 mm × 65 mm in crosssection and 300 mm in length. A honeycomb and settling chamber, together with a contraction nozzle, are used to reduce the turbulence level and make the flow uniform. Velocity measurements show that, except near the endwalls, the velocity is uniform and within 1.8% of the centreline velocity (Kumar, 2012). This ensures a nominally two-dimensional flow field. Independent pressure and velocity control are achieved through the use of a vacuum pump and a circulating water pump, respectively. The test-section pressure can be varied between 0.1 and 1 bar, while the test-section velocity ranges between 4 and 15 m/s. However, since Re affects the wake characteristics, in the work reported here, we used a constant test-section velocity of 6.4 m/s. The test body was a circular cylinder made of brass with a diameter of 10 mm and a surface roughness Ra = 0.6 μm. Thus Re based on cylinder diameter was fixed at 6.4 × 104 . The cylinder spanned from end to end in the test section, giving an aspect ratio of 6.5. The test-section static pressure and the cylinder wall pressure were measured using Honeywell smart pressure sensors (Fig. 1). The cylinder surface pressure data for non-cavitating case were compared with that available in literature and was found to be in good agreement (Kumar et al., 2016). The dynamic pressure on the cylinder wall was measured using a PCB miniature pressure transducer (model 112A22)
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with a resonant frequency of 250 kHz. It may be added here that the Helmholtz resonator frequency corresponding to the cavity in which the transducer was placed was 37.5 kHz and hence did not interfere with the measured signal (Reisman et al., 1998). These data were acquired using an NI DAQ data acquisition system and Labview software. Airborne noise was measured by a sound-level meter (B&K, model 2250) and is expressed in decibel (dB) in the present work with a reference pressure of 20 μ Pa. Dissolved oxygen level and temperature were measured simultaneously using a LDO101 DO probe (Hach, Germany) and were found to be 5.6 ppm and 32 °C, respectively. Measurements in a water sample using an upright microscope (Carl Zeiss Axioscope A1) showed that the size of the largest microparticles and free microbubbles was 62 μm. Uncertainty estimation was carried out following standard practice (Moffat, 1988) and the uncertainties in estimating Re, cavitation number σ and coefficient of pressure Cp were determined to be ± 1%, ± 2.83% and ± 2.83%, respectively. 3. High-speed imaging and image processing The test section had transparent acrylic (Perspex) windows on three sides (Fig. 1), which facilitated imaging of cavitation events. High-speed digital images taken with a Photron Fastcam SA4 camera were used to provide a visual record of cavitation structures. The necessary continuous light source was provided by an LED array. It should be mentioned here that digital high-speed video has been used in a number of recent investigations of cavitating flow, including cavitating flow behind a biconvex hydrofoil (Laberteaux et al., 1998), the cavitation cloud in a convergent–divergent nozzle (Sato et al., 2013) and the void fraction behind a bluff body (Bilus et al., 2014). Aeschlimann et al. (2012), in their work on cavitation in a two-dimensional mixing layer, presented an elaborate technique for extracting useful statistical information from spatiotemporal variation of greyscale intensities. As described in detail below, in the present work, the spatiotemporal variation of image intensities has been used extensively to study cavity dynamics. Images were captured at 360 0 and 10 0 0 frames/s at 1 megapixel resolution in the longitudinal (front) and circular (bottom) views of the cylinder, respectively. A short exposure time of 12 μ s was chosen to provide sharp instantaneous images. For each cavitation number, a high-speed digital movie comprising 10 0 0 images acquired sequentially was used for image processing. Because of backlighting, images of clusters of bubbles appeared as dark cavities in an otherwise bright field. The raw images were filtered using a sliding filter. The idea behind this filter was to retain the sharp regions in the processing, excluding the partially focused regions, which can contribute significantly to noise. An example of this processing is shown in Fig. 2. Fig. 2(a) shows the raw image after scaling in the entire grey level, while Fig. 2(b) shows the filtered image, which retains only those regions that are sufficiently dark and surrounded by similar pixels. The image resolution is 36 μm pixel−1 . Dular et al. (2004), in their work on statistical analysis of cavitation images, indicated that a minimum sample size of 30 images is required. In the present work, we found that a sample size of at least 250 images gave us sufficiently statistically invariant results. The image-processing protocol used to interpret the results is explained here in detail. Image processing was carried out by considering a small region of interest (ROI) and scanned over the entire image in the form of a grid, as represented in Fig. 3(a and b). The size of each cell is 2 pixel by 2 pixel. However, for the sake of clarity representative cells are shown in Fig. 3(a and b) which are not to scale. Each pixel of an image with location (x, z) has intensity level ranging between 0 and 255, with 0 for black pixels and 255 for white. The (x, y, z) axes are shown in Fig. 1. The (x, z) plane is the front view, while the (x, y) plane is the bottom view in the
Fig. 2. Stages of image processing: (a) original image; (b) filtered image.
setup. The mean value of the greyscale of a pixel location (x, z) is given by
g¯ (x, z ) =
N 1 g(x, z, n ), N
(1)
n=1
where N is the number of image frames. Similarly, g(x, y) is also defined for the bottom view. We have reported the mean and other statistical quantities with their normalized (0–1) values as shown for the mean data
G¯ (x, z ) =
g¯ (x, z ) . 255
(2)
The averaged greyscale intensity in a spanwise strip for the front view is defined as
G¯ (x ) =
nz 1 G¯ (x, zk ), nz
(3)
k=1
while the averaged greyscale intensity in a strip for the bottom view is defined as
G¯ (y ) =
nx 1 G¯ (xi , y ), nx
(4)
i=1
The normalized averaged greyscale intensity for the front view is defined as nz nx
G˜ =
G¯ (xi , zk )
k=1 i=1
nx nz
.
(5)
In (3)–(5), nx and nz are the numbers of cells over which the summation is carried out. Greyscale fluctuations, which are indicative of the dynamics of cavity oscillation, are given by
g (x, y, n ) = g(x, y, n ) − g¯ (x, y ),
(6)
with magnitude given by
Grms (x, y ) =
N 1 [g (x, y, n )]2 N n=1
255
.
(7)
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Fig. 3. Grid layout in (a) front view and (b) bottom view. Grid shown in this figure are only indicative of small cells which are too small to be visible. Flow is from right to left.
Fig. 5. Variation of G˜ and sound pressure level (SPL) with cavitation number σ . Fig. 4. Comparison of pressure coefficient of non-cavitating flow (σ /σ i =1.06) with literature.
4. Results and discussion 4.1. Front view of cavity behind cylinder The pressure distribution around the cylinder for non-cavitating flow is presented in Fig. 4 which serves as a validation for the experimental setup used for the present study. As discussed earlier, the test-body is a circular cylinder with 10 mm diameter and hence the blockage in the present set up is 16%. The pressure coefficient is sensitive to blockage of test-section and aspect ratio of cylinder (West and Apelt, 1982). In Fig. 4, we compare present experimental results with that of Weidman (1968) for similar blockage and aspect ratio. Our results are in good agreement with that in this work as can be seen from Fig. 4. It may be noted that the Reynolds number for the case of Weidman (1968) is more than ours. However, as mentioned in the book of Zdravkovich (1997) in the upper subcritical flow regime (TRSL3), effect of Reynolds number on the pressure coefficient is not significant. Fig. 4 also presents results obtained by Hinsberg (2015) and Perrin et al. (2006) corresponding to blockage of 10% and 20% respectively. It can be seen from this figure that the base pressure coefficient reduces with an increase in the blockage. This trend
is also in agreement with the existing literature (West and Apelt, 1982). In the present work, cavitation was detected from visual observation of cavitation structures, as well as through cavitationinduced noise in a method similar to that of Nagaya et al. (2011). In Fig. 5, we present the determination of cavitation inception from the changes in the sound pressure level (SPL), which was corroborated by the visual detection of cavities from high-speed imaging. In this figure and throughout this paper, the cavitation number σ is defined as
σ=
PT − Pv 1 2
ρV 2
,
(8)
where PT is the test-section pressure, Pv is the vapour pressure at room temperature and V is the average flow velocity in the test section. It is clear from this figure that for σ ≥ 3.72, noise does not vary significantly, while there is a sharp increase in noise level at cavitation numbers less than 3.72. For the rest of this paper, the cavitation number is referenced with respect to this inception value σi = 3.72 and is represented as σ /σ i . Fig. 5 also shows that the noise level reaches a maximum and then falls with a further reduction in cavitation number. This has also been observed by other researchers (e.g. Varga and Sebestyen, 1966). The figure also shows the normalized average greyscale intensity G˜ in the domain. An increase in this value with a reduction in cavitation number signifies the appearance of more cavities behind the cylinder.
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Fig. 6. Snapshot of the front view of a cavitating wake, indicating the extent of cavitation at different values of the cavitation number (σ /σ i ):(a) 0.85, (b) 0.80, (c) 0.68, (d) 0.58, (e) 0.54 and (f) 0.50. The white dash line shows the cylinder. The flow is from right to left.
Unlike the SPL, this value indicates a monotonic increase in cavitation with decreasing cavitation number. This suggests that the sound level could have been reduced by cavity–cavity interference, as suggested by Arakeri and Shanmuganathan (1985). The reduction in noise level, especially the high-frequency content, could also be partially attributed to the attenuation of sound in the bubbly medium (Medwin, 1977; Sarkar et al., 2005). These large cavities, formed as a result of lowering of the test-section pressure, can potentially interact with the shedding process. Typical snapshots of the cavitation structures at different cavitation numbers as marked in Fig. 5 (a–f) are shown in Fig. 6. These images were taken on the central plane passing through the axis of the cylinder. This figure shows cavitation in the wake, near the cylinder, and, with decreasing cavitation number, the cavities grow larger and come closer to the cylinder surface. The number of cavities proliferates all along the span, with threadlike filaments indicative of the growth of bubbles in vortex cores. Fig. 7 shows the contour map of the mean and the magnitude of fluctuation of the greyscale intensity along the spanwise direction of the cylinder for different cavitation numbers. The flow is from right to left same as front view. Apart from the fact that the magnitudes of the mean and the fluctuations increase with decreasing cavitation number, a variation in the spanwise direction is also visible from this figure. This feature could be a result of spanwise variation of the vortex structure as reported in the literature for non-cavitating flow (Szepessy, 1994; Szepessy and Bearman, 1992). The cavitating flow bears a clear signature of this variation in terms of both the cavitating region and activity as represented respectively by the mean G¯ (x, y ) and the magnitude of fluctuation Grms (x, y ) of the greyscale intensity function. It can also be seen that the region of cavitation is the same as the region of activity, suggesting that the cavity dynamics is dominant in the region where the cavities are located. In other words, convec-
tion of cavities in the separated region may be taking place over a much longer timescale. Fig. 8 depicts the span-averaged variation of greyscale G¯ (x ) along the streamwise direction for different cavitation numbers. It shows enhancement of the cavitating region in the near wake with decreasing cavitation number. The peak of the average greyscale intensity of the cavitating region in the wake gradually shifts away from the cylinder, while the cavities move closer, as shown in the snapshot in Fig. 6. With the lowering of cavitation number, the cylinder base pressure reduces and hence cavities start appearing at locations closer to the cylinder. However, at the same time the length of the cavity increases. Cavitation activity and hence the peak of it moves away from the cylinder. It can also be observed from Fig. 8 that at x/d ࣃ 2, for all cavitation numbers, there is a significant reduction in cavity size that corresponds to the cavity collapse behaviour observed from the cavity life in high-speed images. Thus this collapse region seems to agree well with the peak in the noise spectrum at x/d = 2 observed by Fry (1984). To probe into cavity dynamics further, a frequency-domain analysis was performed to obtain the dominant frequencies of the cavitation activity. The fast Fourier transform (FFT) of the time-series data of the intensity function g(x, y, n) at three different locations (p, q, r) marked in Fig. 7(d ) is shown in Fig. 9. Of these three points, p corresponds to the region showing a greater presence of cavities and cavitation activity, while q and r are regions that exhibit lower cavitation activity. A peak frequency of about 250 Hz is observed for point p. It is known that Strouhal number measurement is usually based on temporal variation of lift due to vortex shedding from both sides. This has also been suggested by Saito and Sato (2003). Hence, in this view of the image, the corresponding frequency should be 125 Hz. This corresponds to a Strouhal number of 0.2 based on cylinder diameter and average test-section velocity. Hence this frequency represents the large-scale vortex-shedding frequency corresponding to
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Fig. 7. Contour of greyscale intensity of G¯ (LHS) and G (RHS) for the front view of the cavitating wake for different cavitation numbers (σ /σ i ), (a, a ): 0.68, (b, b ): 0.58, (c, c ): 0.54, (d, d ): 0.5. The hatched region shows the cylinder portion. The flow is from right to left.
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Fig. 8. Variation of spanwise-averaged image intensity G¯ (x ) in the streamwise direction for different cavitation numbers.
Fig. 10. Magnitude squared coherence (msc) at f 0 = 250 Hz, for different cavitation numbers σ /σi = 0.85, 0.8, 0.68, 0.54 and 0.5. Here zk = zk − z1 .
ing in the low frequency modulation. It can be noted that similar low frequency sub-harmonics were observed by Kiya and Matsumura (1988) in the near wake of a flat plate placed normal to the non-cavitating flow. Low frequency unsteadiness for non-cavitating flows particularly in the near wake of a cylinder has been reported by Miau and Wang (1999) and Lehmkuhl et al. (2013). In order to investigate this spanwise variation of signal more quantitatively, the magnitude squared coherence (msc), which is a well-known measure of the relationship between two time series data, is used. In the present case, msc (C1k (f)) based on the value of the greyscale intensity was computed as follows:
C1k ( f ) =
|g1k ( f )|2 , g11 ( f )gkk ( f )
(9)
Fig. 9. Frequency spectrum of greyscale level variation at the different points shown in Fig. 7 for σ /σi = 0.50: ( p) x/d = 1, z/d = 3; (q ) x/d = 1, z/d = 2; (r ) x/d = 2, z/d = 2.
the Reynolds number of the flow condition. However, the FFT analysis also reveals the presence of multiple frequencies. The broad band of frequencies ranging from very low to over 1 kHz (for q and r) is an indicator of the interaction of turbulent pressure fluctuations with cavitating structures. It was found from the repeated experiments that the dynamics of cavities reveal the presence of lower frequencies as sub-harmonics of the shedding frequencies in a definite pattern (17 Hz, 35 Hz, 64 Hz ... ). This occurrence could be because the cavities respond to the unsteadiness associated with small eddies present in the wake and thereby result-
Fig. 11. Variation of correlation length with cavitation number. The dotted and dash–dotted lines refer to the range of values of the correlation length for noncavitating flows at similar Re (Szepessy, 1994; Szepessy and Bearman, 1992). The measurement is done at dx = 1 and for different distances along the spanwise direction.
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Fig. 12. Typical snapshot of the circular/end view of a cavitating wake at σ /σi = 0.50. x0 is the distance of the strip (which has width x) from the centre of the cylinder.
where g11 (f) and gkk (f) are the power-spectral densities of points 1 (x/d = 1 at z/d = 0.14) and k (varying along the spanwise direction) and g1k (f) is the cross power-spectral density. The variation of magnitude of msc corresponding to a frequency of about 250 Hz, for varying separation distances and for different cavitation numbers, is plotted in Fig. 10, which shows that the intensities are well correlated for large distances at higher cavitation number (σ /σi = 0.85), whereas the correlation breaks down within a shorter length for lower cavitation numbers. Considering 0.5 as the minimum value up to which correlation may be assumed to be valid, similar to Szepessy and Bearman (1992), a correlation length ( Ldc ) is obtained. This is plotted against cavitation number in Fig. 11, which also shows the correlation lengths for non-cavitating flow at similar values of Re (Szepessy and Bearman, 1992). The lack of correlation at lower cavitation numbers at distances greater than the cylinder diameter is again evident here. This lack of correlation may be attributed to the formation and growth of uncorrelated cavities in the three-dimensional flow field. There could be a dependence of correlation length on the distance from the cylinder (x/d). However, the correlation lengths in the present experiments are measured from the intensity level of the images and it fails in places where the cavitation activity, and hence the signal, is less.
Fig. 13. Contour of greyscale intensity of G¯ (LHS) and G (RHS) for the bottom view of a cavitating wake for different cavitation numbers (σ /σ i ), (a, a ): 0.68, (b, b ): 0.58, (c, c ): 0.54, (d, d ): 0.5. The hatched region shows the cylinder portion. The flow is from right to left.
P. Kumar et al. / International Journal of Multiphase Flow 89 (2017) 207–217
Fig. 14. Variation of average intensity G¯ (y ) along the width of the wake at x/d = 0.5 for different cavitation numbers.
Fig. 15. Contour of the peak spectrum level of the greyscale computed frequency of the cylinder, showing the spread of the shear layer, the formation region and the wake. The flow is from right to left and σ /σi = 0.50.
The rationale for selecting x/d =1 in Fig. 11 is the peak in activity level here (as seen in Fig. 7). 4.2. Bottom view of cavity behind cylinder Fig. 12 shows a typical image of the circular edge of the cylinder for a low cavitation number σ /σi = 0.50. As the cylinder axis is oriented vertically, this view is available from the bottom window of the test section. A front-coated mirror placed below the window helps to visualize the process in the high-speed camera. In this arrangement, adequate visibility was possible at a relatively lower frame speed of the camera (10 0 0 frames s−1 ). Fig. 13 shows the time-averaged greyscale intensity and its fluctuations in this view. As the cavitation number decreases, the cavity moves towards the cylinder and the wake and the shear layer become wider. Finally, cavitation reaches the rearward base of the cylinder. This can be explained from Fig. 14, which shows the time-averaged greyscale intensity profile G¯ (y ) along a strip of thickness 0.18 mm perpendicular to the flow direction and 5 mm away from the centre of the cylinder. At a higher cavitation number σ /σi = 0.85, the cavitation activity is mainly in the shear layer and gradually shifts towards the centre of the wake until finally, at a cavitation number σ /σi = 0.50, the two cavities in the shear layer and the separatedflow region merge, showing a single peak at the centre. Fig. 15 is a filled contour plot of the maximum amplitude of the frequency spectrum at different points and is obtained as follows. The entire region is scanned and the amplitude corresponding to
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the dominant frequency at each point is obtained using FFT at that point. On the basis of this figure, three regions can be identified. The outer region is characterized by an insignificant amplitude of the frequency spectrum. The corresponding signal in the wake also has lower amplitude, except near the rearward base of the cylinder, where a high-amplitude signal is noted. Between these two layers, there lies a shear layer that exhibits a higher amplitude. In order to reveal the frequency content in different regions of Fig. 15 more clearly, we present representative frequency spectra at a few points in the flow. Fig. 16 shows frequency spectra for three points (a–c in Fig. 15) along a vertical line and for three points (d–f in Fig. 15) along the centreline. It can be seen from Fig. 16 that as we move from point a to c through b, a distinct frequency (corresponding to vortex shedding) can only be found at b, indicating extensive cavitation activity, associated with large-scale vortex dynamics, in the shear layer. However, signals from both the outer layer and the wake indicate the presence of a low magnitude of cavitation activity, with a wide range of frequencies characteristics of the background turbulent flow and noise. At point d, close to the base of the cylinder, a very high signal with frequency significantly lower than the vortex-shedding frequency can be observed. This corresponds to the cavities moving very slowly within the low-pressure recirculating region, similar to the conclusion drawn from Fig. 7. As we move from d to f (Fig. 15), a distinct frequency can only be identified at f, which lies in the extended shear layer. Beyond the location where the shear layer intersects the centreline (corresponding to point f), there is a dominant frequency at twice the vortex-shedding frequency, representing the vortex shed from either side of the cylinder. The doubling of frequency in the merged shear layer can be utilized in finding the formation length (Lf ), i.e. the distance from the centre of the cylinder to the shear-layer crossover point. Fig. 17 shows that Lf increases with a reduction in σ /σ i , which may be a consequence of the growth in cavity size in the wake. This organization of cavitating structure in the subcritical flow regime seen here is consistent with the flow structure observed by Maekawa and Mizuno (1967) at a similar Re. To compare the frequency-domain information obtained from the high-speed images with other independent measurements, dynamic pressure measurements were performed at the rearward base of the cylinder. Pressure fluctuations arise due to cavitating and non-cavitating flow conditions. In order to bring out the influence of cavitation on pressure fluctuation over and above the non-cavitating flow, we have normalized the two signals as normalized amplitude (20log P Pcav (ω(ω) ) ). The data at different frequennoncav cies are plotted in Fig. 18. The dominant frequency at lower cavitation number (σ /σi = 0.50) here is the shedding frequency at 253 Hz, which matches closely with that obtained from the highspeed images. This shows that the frequency information from cavitating structures corroborates well with independent measurements of dynamic pressure fluctuations. It can be also noted from Fig. 18 that there is a reduction in the shedding frequency with a reduction in cavitation number. This can be explained in terms of the formation length. As pointed out by Gerrard (1966) and Gnanaskandan and Mahesh (2016), the increase in formation length is accompanied with a reduction in the shedding frequency. With the lowering of cavitation number the formation length increases (Fig. 17) and so the vortex shedding frequency reduces. 5. Conclusions This paper has presented results on cavitation in the near wake of a cylinder. To the best of our knowledge, this is the first detailed analysis of cavitating flow structures in the shear layer and the wake of a cylinder. We have established that at the subcritical Reynolds number, cavities originate primarily in the free shear
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Fig. 16. Frequency spectrum of time series of greyscale level g(x, y, n) at different points as shown in Fig. 15 for σ /σi = 0.50: (a ) x/d = 1, y/d = 0.9; (b) x/d = 1, y/d = 0.45; (c ) x/d = 1, y/d = 0; (d ) x/d = 0.6, y/d = 0; (e ) x/d = 1.75, y/d = 0; ( f ) x/d = 2.5, y/d = 0.
layer but in neither the wake nor the attached boundary layer of the cylinder. Spanwise variation of cavitation activity is evident, with a low-activity region surrounded by high-cavitationfluctuation regions on either side in the spanwise direction. The correlation length decreases from about 3.5 at higher cavitation numbers to about 1 at lower cavitation numbers. On the basis of the magnitude of the frequency signature, it is possible to identify three different regimes with distinct frequency content: in the shear layer, the vortex-shedding frequency is observed, while broadband signals are noted in the wake and the outer layer. These three zones resemble the zones defined by Maekawa and Mizuno (1967) on the basis of velocity fluctuations measured in a noncavitating flow. The frequencies from the measured pressure fluctuations on the rearward side of the cylinder correlate well with the frequencies obtained from images. Acknowledgement The authors acknowledge the partial financial support of this work by the Naval Research Board of India.
Fig. 17. Variation of formation length (Lf ) with cavitation number.
Fig. 18. Frequency spectrum of pressure fluctuations on the cylinder surface for different cavitation numbers.
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References Aeschlimann, V., Prothin, S., Barre, S., Djeridi, H., 2012. High speed visualizations of the cavitating vortices of 2d mixing layer. Eur. J. Mech. B/Fluids 31, 171–180. Arakeri, V.H., Shanmuganathan, V., 1985. On the evidence for the bubble interference on cavitation noise. J. Fluid Mech. 159, 131–150. Bilus, I., Bombek, G., Hocevar, M., Sirok, B., Cencic, T., Petkovsek, M., 2014. The experimental analysis of cavitating structure fluctuations and pressure pulsations in the cavitation station. J. Mech. Eng. 60(3), 147–157. Bloor, M.S., 1964. The transition to turbulence in the wake of a circular cylinder. J. Fluid Mech. 19, 290–304. Bloor, M.S., Gerrard, J.H., 1966. Measurements on turbulent vortices in a cylinder wake. Proc. Royal. Soc. Lond. A 319–342. Dular, M., Bachert, B., Stoffel, B., Sirok, B., 2004. Relationship between cavitation structures and cavitation damage. Wear 257, 1176–1184. Fry, S.A., 1984. Investigating cavity/wake dynamics for a circular cylinder by measuring noise spectra. J. Fluid Mech. 42, 187–200. Gerrard, J.H., 1966. Mechanics of the formation region of vortices behind bluff bodies. J . Fluid Mech. 25, 401–413. Gnanaskandan, A., Mahesh, K., 2016. Numerical investigation of near-wake characteristics of cavitating flow over a circular cylinder. J. Fluid Mech. 790, 453–491. Griffin, O.M., Votaw, C.W., 1972. The vortex street in the wake of a vibrating cylinder. J. Fluid Mech. 51, 31–48. Hinsberg, N.P.H., 2015. The reynolds number dependency of the steady and unsteady loading on a slightly rough circular cylinder :from subcritical up to high transcritical flow state. J. Fluids Struc. 55, 526–539. Kiya, M., Matsumura, M., 1988. Incoherent turbulence structure in the near wake of a normal plate. J. Fluid Mech. 190, 343–356. Kumar, P., Chatterjee, D., Bakshi, S., 2016. Experimental investigation of cavitation behind a circular cylinder in cross-flow. In: 9th International Conference on Thermal Engineering: Theory and Applications, March 24–26, Abu Dhabi, UAE. Kumar, T.M.P., 2012. Experimental and Numerical Investigation of Non-Cavitating and Cavitating Flows over S-shaped Hydrofoil. Ph.D. thesis. Indian Institute of Technology, Madras, India. Kumar, T.M.P., Kumar, P., Chatterjee, D., 2014. Cavitation characteristics of s-blade used in fully reversible pump-turbine. ASME J. Fluids Eng. 136(5), 1–15. Laberteaux, K.R., Ceccio, S.L., Mastrocola, V.J., Lowrance, J.L., 1998. High speed digital imaging of cavitating vortices. Expt. Fluids. 24, 489–498. Lehmkuhl, O., Rodrguez, I., Borrell, R., Chiva, J., Oliva, A., 2013. Low-frequency unsteadiness in the vortex formation region of a circular cylinder. Phys. Fluids 25, 085109. Linke, W., 1931. Neue messungen zur aerodynamik des zylinders. insbesondere seines reinen Reibungswiderstandes. Phys. Zr. 32, 900. Maekawa, T., Mizuno, S., 1967. Flow around the separation point and in the nearwake of a circular cylinder. Phys. Fluids 10, S184–S186. Matsudaira, Y., Gomi, Y., Oba, R., 1992. Characteristics of bubble-collapse pressures in a karman-vortex cavity. JSME INT. J., Ser. II. 35, 179–185. Medwin, H., 1977. Counting bubbles acoustically: a review. Ultrasonics. 15, 7–13. Miau, J., Wang, J.T., 1999. Characteristics of low-frequency variations embedded in vortex-shedding process. J. Fluids and Struc. 13, 339–359. Moffat, R.J., 1988. Describing the uncertainties in experimental results. Expt. Thermal Fluid Sci. 3–17. Nagaya, Y., Murase, M., Mizuyama, S., 2011. Detection of cavitation with microphone placed outside piping. J. of Environ. Eng. 6, 426–436.
217
Norberg, C., 1994. An experimental investigation of the flow around a circular cylinder: influence of aspect ratio. J. Fluid Mech. 258, 287–316. Perrin, R., Braza, M., Cid, E., Cazin, S., Moradei, F., Barthet, A., Sevrain, A., Hoarau, Y., 2006. Near-wake turbulence properties in the high reynolds number incompressible flow around a circular cylinder measured by two and three-component piv. Flow Turbulence Combust 77, 185–204. Ramamurthy, A., Bhaskaran, P., 1977. Constrained flow past cavitating bluff bodies. ASME J. Fluids Eng. 99, 717–726. Rao, B.C.S., Chandrasekhara, D.V., 1976. Some characteristics of cavity flow past cylindrical inducers in a venturi. ASME J. Fluids Eng. 98, 461–468.. Reisman, G.E., Wang, Y.C., Brennen, C.E., 1998. Observations of shock waves in cloud cavitation. J. Fluid Mech. 355, 255–283.. Roshko, A., 1954. On the development of turbulent wakes from vortex streets. NACA Rep. 1191. Roshko, A., 1961. Experiments on the flow past a circular cylinder at very high reynolds number. J. Fluid Mech. 10, 345–356. Roshko, A., 1993. Perspectives on bluff body aerodynamics. J. Wind Eng. Indust. Aerodyn. 49, 79. Saito, Y.A., Sato, K., 2003. Cavitation bubble collapse and impact in the wake of circular cylinder. Fifth International Symposium on Cavitation (cav2003), Osaka, Japan. Sarkar, K., Shi, W.T., Chatterjee, D., Forsberg, F., 2005. Characterization of ultrasound contrast microbubbles using in vitro experiments and viscous and viscoelastic interface models for encapsulation.. J. Acoust. Soc. Am. 118, 539–550. Sato, K., Taguchi, Y., Hayashi, S., 2013. High speed observation of periodic cavity behavior in a convergent-divergent nozzle for cavitating water jet. J. of Flow Control, Meas. Visual. 1, 102–107. Schewe, G., 1983. On the force fluctuations acting on a circular cylinder in cross-flow from subcritical up to transcritical reynolds numbers. J Fluid Mech. 133, 265–285. Shalnev, K., 1965. Boundary effect on cavitating flow past a cylinder. J. Appl. Mech. Tech. Phys. 3, 103–108. Szepessy, S., 1994. On the spanwise correlation of vortex shedding from a circular cylinder at high subcritical reynolds number. Phys. Fluids 6, 2406–2416. Szepessy, S., Bearman, P., 1992. Aspect ratio and end plate effects on vortex shedding from a circular cylinder. J. Fluid Mech. 234, 191–217. Unal, M.F., Rockwell, D., 1988. On the vortex formation from a cylinder part 1. the initial instability. J. Fluid Mech. 190, 491–512. Varga, J., Sebestyen, G., 1966. Experimental investigation of cavitation noise. Houille Blanche 8, 905–910. Wei, T., Smith, C.R., 1986. Secondary vortices in the wake of circular cylinders. J. Fluid Mech. 169, 513–533. Weidman, P.D., 1968. Wake Transition and Blockage Effects on Cylinder Base Pressures. Ph.D. thesis. California Institute of Technology, Pasadena, California. West, G.S., Apelt, C.J., 1982. The effects of tunnel blockage and aspect ratio on the mean flow past a circular cylinder with reynolds numbers between 104 and 105 . J. Fluid Mech. 114, 361–377. Williamson, C.H.K., 1996. Three-dimensional wake transition. J. Fluid Mech. 328, 345–407. Zdravkovich, M.M., 1997. Flow around Circular Cylinders. Fundamentals. Oxford University Press, New York.
Contents lists available at ScienceDirect
International Journal of Multiphase Flow journal homepage: www.elsevier.com/locate/ijmulflow
Experimental investigation of cavitating structures in the near wake of a cylinder Pankaj Kumar, Dhiman Chatterjee∗, Shamit Bakshi Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai 600036, India
a r t i c l e
i n f o
Article history: Received 19 February 2016 Revised 12 August 2016 Accepted 24 September 2016 Available online 11 November 2016 Keywords: Cavitation High speed images Shear layer Formation length
a b s t r a c t An experimental investigation of cavitating structures in the near-wake region of a cylinder is presented. From high-speed imaging of this subcritical flow (Reynolds number of 64,0 0 0), it is found that inception of cavities occurs in the shear layer. At the developed cavitation condition, the cavities in the separated zone and the free shear layer merge. A distinct spanwise variation in cavitation activity is observed. The non-dimensionalized correlation length at inception varies from close to a non-cavitating value of about 3.5 to about 1 at developed cavitation. The non-dimensionalized length of formation, characterized by crossover of the free shear layer and the wake axis, increases from 1 to 1.8 as the cavitation number is reduced from 85% to 50% of the inception value. A frequency analysis of the cavity dynamics indicates that although the vortex shedding frequency is dominant in the shear layer, there are peaks corresponding to other frequencies in other flow regions. The presence of a sharp peak at 125 Hz, corresponding to a Strouhal number of 0.2, along with a range of frequencies, is also verified independently through measurement of fluctuating pressure at the cylinder surface. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction Non-cavitating flow past a circular cylinder has been the subject of extensive research, since this simple geometry exhibits interesting fluid phenomena and also has wide engineering applications. This flow is classified as subcritical, critical or supercritical, depending on the value of the Reynolds number Re (Roshko, 1961; Schewe, 1983; Zdravkovich, 1997), with each flow regime exhibiting distinctive features, such as the location of the transition to turbulence (in the wake, shear layer or boundary layer) and different characteristic frequencies of pressure and velocity fluctuations. A review of the literature reveals that most investigations have been carried out at low Re and have been directed at understanding two-dimensional vortex shedding as well as the origin and growth of three-dimensional disturbances, which can be important even for nominally two-dimensional cylinders (Roshko, 1993; Williamson, 1996). The transition in the wake takes place at much lower Re. Distinct irregularities in velocity fluctuations becomes visible as Re exceeds 300 (Roshko, 1954), which Roshko (1954) characterized as the irregular regime. Bloor (1964) observed the first transition eddies at Re = 1300 and found the frequency of the transition eddies to scale as Re0.5 . Wei and Smith (1986) noted
∗
Corresponding author. Fax: +9144 22574652. E-mail address: [email protected] (D. Chatterjee).
http://dx.doi.org/10.1016/j.ijmultiphaseflow.2016.09.025 0301-9322/© 2016 Elsevier Ltd. All rights reserved.
that the formation of ‘secondary vortices’ was linked with these high-frequency fluctuations and their results showed an Re0.77 dependence. As transition proceeds to the shear layer, the coefficient of base suction pressure (−C pb ) increases, the Reynolds stress level increases and the Strouhal number gradually decreases (Norberg, 1994). With further increase in Re in the upper subcritical regime (2 × 104 < Re < 2 × 105 ), the flow features change. There have been fewer studies of this regime, which is briefly described in the next paragraph. One important characteristic of the wake is the eddy-formation region. This region has been defined differently by different researchers: as the region behind the cylinder up to the minimum of the measured Cp distribution along the wake axis (Linke, 1931), as the region where low-frequency oscillations disappear (Bloor, 1964), as the location at which the free shear layer crosses the wake axis (Bloor and Gerrard, 1966) and as the location of maximum fluctuations umax in the wake (Griffin and Votaw, 1972). The formation length (Lf ) is the distance, along the flow direction, of the eddy-formation region from the cylinder centre. This formation length (Lf ), when non-dimensionalized with the cylinder diameter (d), decreases with increasing Re in the range Re = 103 –2 × 105 and is typically less than 2 above Re = 10 0 0 0. The aspect ratio L/D of the cylinder, besides the Re of the flow, also influences this value. This shortening of Lf , coupled with a widening of the shear layer at higher Re, was first observed by Gerrard (1966). Unal and Rockwell (1988) used a flow visualization technique to reveal the
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decrease in Lf . Further increase in Re does not lead to any significant shortening of Lf . The overall organization of the flow regimes behind a cylinder in the subcritical regime was nicely shown in an experiment by Maekawa and Mizuno (1967), who used hot-wire signals to detect the free shear layer and the near-wake characteristics at Re = 65 × 103 . It was found that the signals detected in the near wake are random low-frequency velocity fluctuations, while those from the free shear layer exhibit the vortex-shedding frequency. In the region downstream of the formation region along the axis, there are two components to the signal, which signifies alternating crossing of flows from the two sides of the wake. This feature is probed further in the present work in the context of cavitating flow. The flow is inherently and highly three-dimensional, but, as indicated by Szepessy and Bearman (1992), for very small L/D values, nominally ‘two-dimensional’ conditions can be assumed. The correlation length, i.e. the length within which the pressure signals are highly correlated (with correlation coefficient ≥ 0.5) along the spanwise direction, was found to be 2.5 times the cylinder diameter for Re ࣃ 71 × 103 (Szepessy and Bearman, 1992) and 3.5 times the cylinder diameter for Re ࣃ 43 × 103 (Szepessy, 1994). In the present work, the L/D of the cylinder is similar to that in Szepessy and Bearman (1992), which helps to reveal the influence of the extent of cavitation activity on the correlation length. In comparison with non-cavitating flow, there has been relatively less research on cavitating flow past a circular cylinder. Furthermore, most investigations to date have examined the critical and supercritical flow regimes. A survey of the literature on cavitating flow past a circular cylinder shows that the major thrust of research has been on determining variations in overall cavity geometry (length and width) and drag coefficient as functions of cavitation number. For example, in one of the earliest pieces of work in this area, Shalnev (1965) obtained the cavity length and width and the drag coefficient as functions of the blockage factor in a constrained test section. The significance of the blockage factor was also discussed by Ramamurthy and Bhaskaran (1977) and Rao and Chandrasekhara (1976). The effects of cavitation number on cavity length and width as well as on vortex-shedding frequency were studied by Varga and Sebestyen (1966) for supercritical flow. Rao and Chandrasekhara (1976), on the basis of highspeed movies, noted that there were considerable longitudinal oscillations downstream of the cavity. This finding highlights the importance of characterizing local cavity oscillations in order to understand the interaction of vortices with the cavity. In a related study of local dynamics, Fry (1984) determined the cavitation noise spectrum at different downstream locations in the wake for different cavitation numbers. His results indicate that the peak noise was observed when the cavity length was twice the cylinder diameter. Matsudaira et al. (1992) measured the dynamic pressure due to bubble collapse at several locations in the wake of a cylinder for an Re range of 4.5 × 105 –6 × 105 . In the supercritical flow regime, they found that the frequency of peaks due to impulsive pressure was high at x/d ࣃ 1.3 in the separation region, while in the vortex-formation region (2.1 < x/d < 2.9), peaks were fewer and were mixed with background turbulent fluctuations. They further found that, with a decrease in cavitation number, the cavity length determined from pressure contour maps increased and extended downstream. Investigation of the local cavity dynamics using high-speed imaging forms an important part of the present work. It is clear from the literature cited above that there has been a few investigations of cavitating flow past a cylinder in the upper subcritical flow region (20, 0 0 0 < Re < 20 0, 0 0 0). Studies of noncavitating flow in this regime have led to interesting observations such as the organization of flow regimes (Maekawa and Mizuno, 1967), the phase drift and the associated three-dimensionality be-
Fig. 1. Schematic plan view of the test section with instrument. The inset shows the front view of the cylinder inside the test section. Also shown is the (x, y, z) axis system used in this paper. The origin is at the centre of the cylinder and is aligned with the lower edge of the test-section window.
yond a certain spanwise distance (Szepessy, 1994; Szepessy and Bearman, 1992). Hence the pertinent question that needs to be addressed is the way in which these observations may be modified as we move gradually from non-cavitating through weakly cavitating to a highly cavitating regime at the same value of Re. Furthermore, most work, even in the critical or supercritical regime, has been related to measurements of cylinder surface pressure and drag, overall cavity structure, and cavitation-induced noise, and has not dealt with cavity dynamics. Following Fry (1984), we suggest that although cavity length and cavity-shedding frequency have already been measured behind cavitating circular cylinders, these cannot explain the location of the peak noise value. Hence, in this work, we attempt to gain further insight by presenting results that relate the dynamics of cavitating structures and the frequency of fluctuation of cavities to the cavitation number. In Section 2, the experimental facility is described. This is followed by a description of the image processing in Section 3 and the results are then discussed in Section 4. Our conclusions are given in Section 5. 2. Experimental facility and methodology The cavitation test rig used in the present study is described in detail by Kumar et al. (2014) and in this section we describe only the major aspects. The test section is 65 mm × 65 mm in crosssection and 300 mm in length. A honeycomb and settling chamber, together with a contraction nozzle, are used to reduce the turbulence level and make the flow uniform. Velocity measurements show that, except near the endwalls, the velocity is uniform and within 1.8% of the centreline velocity (Kumar, 2012). This ensures a nominally two-dimensional flow field. Independent pressure and velocity control are achieved through the use of a vacuum pump and a circulating water pump, respectively. The test-section pressure can be varied between 0.1 and 1 bar, while the test-section velocity ranges between 4 and 15 m/s. However, since Re affects the wake characteristics, in the work reported here, we used a constant test-section velocity of 6.4 m/s. The test body was a circular cylinder made of brass with a diameter of 10 mm and a surface roughness Ra = 0.6 μm. Thus Re based on cylinder diameter was fixed at 6.4 × 104 . The cylinder spanned from end to end in the test section, giving an aspect ratio of 6.5. The test-section static pressure and the cylinder wall pressure were measured using Honeywell smart pressure sensors (Fig. 1). The cylinder surface pressure data for non-cavitating case were compared with that available in literature and was found to be in good agreement (Kumar et al., 2016). The dynamic pressure on the cylinder wall was measured using a PCB miniature pressure transducer (model 112A22)
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with a resonant frequency of 250 kHz. It may be added here that the Helmholtz resonator frequency corresponding to the cavity in which the transducer was placed was 37.5 kHz and hence did not interfere with the measured signal (Reisman et al., 1998). These data were acquired using an NI DAQ data acquisition system and Labview software. Airborne noise was measured by a sound-level meter (B&K, model 2250) and is expressed in decibel (dB) in the present work with a reference pressure of 20 μ Pa. Dissolved oxygen level and temperature were measured simultaneously using a LDO101 DO probe (Hach, Germany) and were found to be 5.6 ppm and 32 °C, respectively. Measurements in a water sample using an upright microscope (Carl Zeiss Axioscope A1) showed that the size of the largest microparticles and free microbubbles was 62 μm. Uncertainty estimation was carried out following standard practice (Moffat, 1988) and the uncertainties in estimating Re, cavitation number σ and coefficient of pressure Cp were determined to be ± 1%, ± 2.83% and ± 2.83%, respectively. 3. High-speed imaging and image processing The test section had transparent acrylic (Perspex) windows on three sides (Fig. 1), which facilitated imaging of cavitation events. High-speed digital images taken with a Photron Fastcam SA4 camera were used to provide a visual record of cavitation structures. The necessary continuous light source was provided by an LED array. It should be mentioned here that digital high-speed video has been used in a number of recent investigations of cavitating flow, including cavitating flow behind a biconvex hydrofoil (Laberteaux et al., 1998), the cavitation cloud in a convergent–divergent nozzle (Sato et al., 2013) and the void fraction behind a bluff body (Bilus et al., 2014). Aeschlimann et al. (2012), in their work on cavitation in a two-dimensional mixing layer, presented an elaborate technique for extracting useful statistical information from spatiotemporal variation of greyscale intensities. As described in detail below, in the present work, the spatiotemporal variation of image intensities has been used extensively to study cavity dynamics. Images were captured at 360 0 and 10 0 0 frames/s at 1 megapixel resolution in the longitudinal (front) and circular (bottom) views of the cylinder, respectively. A short exposure time of 12 μ s was chosen to provide sharp instantaneous images. For each cavitation number, a high-speed digital movie comprising 10 0 0 images acquired sequentially was used for image processing. Because of backlighting, images of clusters of bubbles appeared as dark cavities in an otherwise bright field. The raw images were filtered using a sliding filter. The idea behind this filter was to retain the sharp regions in the processing, excluding the partially focused regions, which can contribute significantly to noise. An example of this processing is shown in Fig. 2. Fig. 2(a) shows the raw image after scaling in the entire grey level, while Fig. 2(b) shows the filtered image, which retains only those regions that are sufficiently dark and surrounded by similar pixels. The image resolution is 36 μm pixel−1 . Dular et al. (2004), in their work on statistical analysis of cavitation images, indicated that a minimum sample size of 30 images is required. In the present work, we found that a sample size of at least 250 images gave us sufficiently statistically invariant results. The image-processing protocol used to interpret the results is explained here in detail. Image processing was carried out by considering a small region of interest (ROI) and scanned over the entire image in the form of a grid, as represented in Fig. 3(a and b). The size of each cell is 2 pixel by 2 pixel. However, for the sake of clarity representative cells are shown in Fig. 3(a and b) which are not to scale. Each pixel of an image with location (x, z) has intensity level ranging between 0 and 255, with 0 for black pixels and 255 for white. The (x, y, z) axes are shown in Fig. 1. The (x, z) plane is the front view, while the (x, y) plane is the bottom view in the
Fig. 2. Stages of image processing: (a) original image; (b) filtered image.
setup. The mean value of the greyscale of a pixel location (x, z) is given by
g¯ (x, z ) =
N 1 g(x, z, n ), N
(1)
n=1
where N is the number of image frames. Similarly, g(x, y) is also defined for the bottom view. We have reported the mean and other statistical quantities with their normalized (0–1) values as shown for the mean data
G¯ (x, z ) =
g¯ (x, z ) . 255
(2)
The averaged greyscale intensity in a spanwise strip for the front view is defined as
G¯ (x ) =
nz 1 G¯ (x, zk ), nz
(3)
k=1
while the averaged greyscale intensity in a strip for the bottom view is defined as
G¯ (y ) =
nx 1 G¯ (xi , y ), nx
(4)
i=1
The normalized averaged greyscale intensity for the front view is defined as nz nx
G˜ =
G¯ (xi , zk )
k=1 i=1
nx nz
.
(5)
In (3)–(5), nx and nz are the numbers of cells over which the summation is carried out. Greyscale fluctuations, which are indicative of the dynamics of cavity oscillation, are given by
g (x, y, n ) = g(x, y, n ) − g¯ (x, y ),
(6)
with magnitude given by
Grms (x, y ) =
N 1 [g (x, y, n )]2 N n=1
255
.
(7)
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Fig. 3. Grid layout in (a) front view and (b) bottom view. Grid shown in this figure are only indicative of small cells which are too small to be visible. Flow is from right to left.
Fig. 5. Variation of G˜ and sound pressure level (SPL) with cavitation number σ . Fig. 4. Comparison of pressure coefficient of non-cavitating flow (σ /σ i =1.06) with literature.
4. Results and discussion 4.1. Front view of cavity behind cylinder The pressure distribution around the cylinder for non-cavitating flow is presented in Fig. 4 which serves as a validation for the experimental setup used for the present study. As discussed earlier, the test-body is a circular cylinder with 10 mm diameter and hence the blockage in the present set up is 16%. The pressure coefficient is sensitive to blockage of test-section and aspect ratio of cylinder (West and Apelt, 1982). In Fig. 4, we compare present experimental results with that of Weidman (1968) for similar blockage and aspect ratio. Our results are in good agreement with that in this work as can be seen from Fig. 4. It may be noted that the Reynolds number for the case of Weidman (1968) is more than ours. However, as mentioned in the book of Zdravkovich (1997) in the upper subcritical flow regime (TRSL3), effect of Reynolds number on the pressure coefficient is not significant. Fig. 4 also presents results obtained by Hinsberg (2015) and Perrin et al. (2006) corresponding to blockage of 10% and 20% respectively. It can be seen from this figure that the base pressure coefficient reduces with an increase in the blockage. This trend
is also in agreement with the existing literature (West and Apelt, 1982). In the present work, cavitation was detected from visual observation of cavitation structures, as well as through cavitationinduced noise in a method similar to that of Nagaya et al. (2011). In Fig. 5, we present the determination of cavitation inception from the changes in the sound pressure level (SPL), which was corroborated by the visual detection of cavities from high-speed imaging. In this figure and throughout this paper, the cavitation number σ is defined as
σ=
PT − Pv 1 2
ρV 2
,
(8)
where PT is the test-section pressure, Pv is the vapour pressure at room temperature and V is the average flow velocity in the test section. It is clear from this figure that for σ ≥ 3.72, noise does not vary significantly, while there is a sharp increase in noise level at cavitation numbers less than 3.72. For the rest of this paper, the cavitation number is referenced with respect to this inception value σi = 3.72 and is represented as σ /σ i . Fig. 5 also shows that the noise level reaches a maximum and then falls with a further reduction in cavitation number. This has also been observed by other researchers (e.g. Varga and Sebestyen, 1966). The figure also shows the normalized average greyscale intensity G˜ in the domain. An increase in this value with a reduction in cavitation number signifies the appearance of more cavities behind the cylinder.
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Fig. 6. Snapshot of the front view of a cavitating wake, indicating the extent of cavitation at different values of the cavitation number (σ /σ i ):(a) 0.85, (b) 0.80, (c) 0.68, (d) 0.58, (e) 0.54 and (f) 0.50. The white dash line shows the cylinder. The flow is from right to left.
Unlike the SPL, this value indicates a monotonic increase in cavitation with decreasing cavitation number. This suggests that the sound level could have been reduced by cavity–cavity interference, as suggested by Arakeri and Shanmuganathan (1985). The reduction in noise level, especially the high-frequency content, could also be partially attributed to the attenuation of sound in the bubbly medium (Medwin, 1977; Sarkar et al., 2005). These large cavities, formed as a result of lowering of the test-section pressure, can potentially interact with the shedding process. Typical snapshots of the cavitation structures at different cavitation numbers as marked in Fig. 5 (a–f) are shown in Fig. 6. These images were taken on the central plane passing through the axis of the cylinder. This figure shows cavitation in the wake, near the cylinder, and, with decreasing cavitation number, the cavities grow larger and come closer to the cylinder surface. The number of cavities proliferates all along the span, with threadlike filaments indicative of the growth of bubbles in vortex cores. Fig. 7 shows the contour map of the mean and the magnitude of fluctuation of the greyscale intensity along the spanwise direction of the cylinder for different cavitation numbers. The flow is from right to left same as front view. Apart from the fact that the magnitudes of the mean and the fluctuations increase with decreasing cavitation number, a variation in the spanwise direction is also visible from this figure. This feature could be a result of spanwise variation of the vortex structure as reported in the literature for non-cavitating flow (Szepessy, 1994; Szepessy and Bearman, 1992). The cavitating flow bears a clear signature of this variation in terms of both the cavitating region and activity as represented respectively by the mean G¯ (x, y ) and the magnitude of fluctuation Grms (x, y ) of the greyscale intensity function. It can also be seen that the region of cavitation is the same as the region of activity, suggesting that the cavity dynamics is dominant in the region where the cavities are located. In other words, convec-
tion of cavities in the separated region may be taking place over a much longer timescale. Fig. 8 depicts the span-averaged variation of greyscale G¯ (x ) along the streamwise direction for different cavitation numbers. It shows enhancement of the cavitating region in the near wake with decreasing cavitation number. The peak of the average greyscale intensity of the cavitating region in the wake gradually shifts away from the cylinder, while the cavities move closer, as shown in the snapshot in Fig. 6. With the lowering of cavitation number, the cylinder base pressure reduces and hence cavities start appearing at locations closer to the cylinder. However, at the same time the length of the cavity increases. Cavitation activity and hence the peak of it moves away from the cylinder. It can also be observed from Fig. 8 that at x/d ࣃ 2, for all cavitation numbers, there is a significant reduction in cavity size that corresponds to the cavity collapse behaviour observed from the cavity life in high-speed images. Thus this collapse region seems to agree well with the peak in the noise spectrum at x/d = 2 observed by Fry (1984). To probe into cavity dynamics further, a frequency-domain analysis was performed to obtain the dominant frequencies of the cavitation activity. The fast Fourier transform (FFT) of the time-series data of the intensity function g(x, y, n) at three different locations (p, q, r) marked in Fig. 7(d ) is shown in Fig. 9. Of these three points, p corresponds to the region showing a greater presence of cavities and cavitation activity, while q and r are regions that exhibit lower cavitation activity. A peak frequency of about 250 Hz is observed for point p. It is known that Strouhal number measurement is usually based on temporal variation of lift due to vortex shedding from both sides. This has also been suggested by Saito and Sato (2003). Hence, in this view of the image, the corresponding frequency should be 125 Hz. This corresponds to a Strouhal number of 0.2 based on cylinder diameter and average test-section velocity. Hence this frequency represents the large-scale vortex-shedding frequency corresponding to
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Fig. 7. Contour of greyscale intensity of G¯ (LHS) and G (RHS) for the front view of the cavitating wake for different cavitation numbers (σ /σ i ), (a, a ): 0.68, (b, b ): 0.58, (c, c ): 0.54, (d, d ): 0.5. The hatched region shows the cylinder portion. The flow is from right to left.
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Fig. 8. Variation of spanwise-averaged image intensity G¯ (x ) in the streamwise direction for different cavitation numbers.
Fig. 10. Magnitude squared coherence (msc) at f 0 = 250 Hz, for different cavitation numbers σ /σi = 0.85, 0.8, 0.68, 0.54 and 0.5. Here zk = zk − z1 .
ing in the low frequency modulation. It can be noted that similar low frequency sub-harmonics were observed by Kiya and Matsumura (1988) in the near wake of a flat plate placed normal to the non-cavitating flow. Low frequency unsteadiness for non-cavitating flows particularly in the near wake of a cylinder has been reported by Miau and Wang (1999) and Lehmkuhl et al. (2013). In order to investigate this spanwise variation of signal more quantitatively, the magnitude squared coherence (msc), which is a well-known measure of the relationship between two time series data, is used. In the present case, msc (C1k (f)) based on the value of the greyscale intensity was computed as follows:
C1k ( f ) =
|g1k ( f )|2 , g11 ( f )gkk ( f )
(9)
Fig. 9. Frequency spectrum of greyscale level variation at the different points shown in Fig. 7 for σ /σi = 0.50: ( p) x/d = 1, z/d = 3; (q ) x/d = 1, z/d = 2; (r ) x/d = 2, z/d = 2.
the Reynolds number of the flow condition. However, the FFT analysis also reveals the presence of multiple frequencies. The broad band of frequencies ranging from very low to over 1 kHz (for q and r) is an indicator of the interaction of turbulent pressure fluctuations with cavitating structures. It was found from the repeated experiments that the dynamics of cavities reveal the presence of lower frequencies as sub-harmonics of the shedding frequencies in a definite pattern (17 Hz, 35 Hz, 64 Hz ... ). This occurrence could be because the cavities respond to the unsteadiness associated with small eddies present in the wake and thereby result-
Fig. 11. Variation of correlation length with cavitation number. The dotted and dash–dotted lines refer to the range of values of the correlation length for noncavitating flows at similar Re (Szepessy, 1994; Szepessy and Bearman, 1992). The measurement is done at dx = 1 and for different distances along the spanwise direction.
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Fig. 12. Typical snapshot of the circular/end view of a cavitating wake at σ /σi = 0.50. x0 is the distance of the strip (which has width x) from the centre of the cylinder.
where g11 (f) and gkk (f) are the power-spectral densities of points 1 (x/d = 1 at z/d = 0.14) and k (varying along the spanwise direction) and g1k (f) is the cross power-spectral density. The variation of magnitude of msc corresponding to a frequency of about 250 Hz, for varying separation distances and for different cavitation numbers, is plotted in Fig. 10, which shows that the intensities are well correlated for large distances at higher cavitation number (σ /σi = 0.85), whereas the correlation breaks down within a shorter length for lower cavitation numbers. Considering 0.5 as the minimum value up to which correlation may be assumed to be valid, similar to Szepessy and Bearman (1992), a correlation length ( Ldc ) is obtained. This is plotted against cavitation number in Fig. 11, which also shows the correlation lengths for non-cavitating flow at similar values of Re (Szepessy and Bearman, 1992). The lack of correlation at lower cavitation numbers at distances greater than the cylinder diameter is again evident here. This lack of correlation may be attributed to the formation and growth of uncorrelated cavities in the three-dimensional flow field. There could be a dependence of correlation length on the distance from the cylinder (x/d). However, the correlation lengths in the present experiments are measured from the intensity level of the images and it fails in places where the cavitation activity, and hence the signal, is less.
Fig. 13. Contour of greyscale intensity of G¯ (LHS) and G (RHS) for the bottom view of a cavitating wake for different cavitation numbers (σ /σ i ), (a, a ): 0.68, (b, b ): 0.58, (c, c ): 0.54, (d, d ): 0.5. The hatched region shows the cylinder portion. The flow is from right to left.
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Fig. 14. Variation of average intensity G¯ (y ) along the width of the wake at x/d = 0.5 for different cavitation numbers.
Fig. 15. Contour of the peak spectrum level of the greyscale computed frequency of the cylinder, showing the spread of the shear layer, the formation region and the wake. The flow is from right to left and σ /σi = 0.50.
The rationale for selecting x/d =1 in Fig. 11 is the peak in activity level here (as seen in Fig. 7). 4.2. Bottom view of cavity behind cylinder Fig. 12 shows a typical image of the circular edge of the cylinder for a low cavitation number σ /σi = 0.50. As the cylinder axis is oriented vertically, this view is available from the bottom window of the test section. A front-coated mirror placed below the window helps to visualize the process in the high-speed camera. In this arrangement, adequate visibility was possible at a relatively lower frame speed of the camera (10 0 0 frames s−1 ). Fig. 13 shows the time-averaged greyscale intensity and its fluctuations in this view. As the cavitation number decreases, the cavity moves towards the cylinder and the wake and the shear layer become wider. Finally, cavitation reaches the rearward base of the cylinder. This can be explained from Fig. 14, which shows the time-averaged greyscale intensity profile G¯ (y ) along a strip of thickness 0.18 mm perpendicular to the flow direction and 5 mm away from the centre of the cylinder. At a higher cavitation number σ /σi = 0.85, the cavitation activity is mainly in the shear layer and gradually shifts towards the centre of the wake until finally, at a cavitation number σ /σi = 0.50, the two cavities in the shear layer and the separatedflow region merge, showing a single peak at the centre. Fig. 15 is a filled contour plot of the maximum amplitude of the frequency spectrum at different points and is obtained as follows. The entire region is scanned and the amplitude corresponding to
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the dominant frequency at each point is obtained using FFT at that point. On the basis of this figure, three regions can be identified. The outer region is characterized by an insignificant amplitude of the frequency spectrum. The corresponding signal in the wake also has lower amplitude, except near the rearward base of the cylinder, where a high-amplitude signal is noted. Between these two layers, there lies a shear layer that exhibits a higher amplitude. In order to reveal the frequency content in different regions of Fig. 15 more clearly, we present representative frequency spectra at a few points in the flow. Fig. 16 shows frequency spectra for three points (a–c in Fig. 15) along a vertical line and for three points (d–f in Fig. 15) along the centreline. It can be seen from Fig. 16 that as we move from point a to c through b, a distinct frequency (corresponding to vortex shedding) can only be found at b, indicating extensive cavitation activity, associated with large-scale vortex dynamics, in the shear layer. However, signals from both the outer layer and the wake indicate the presence of a low magnitude of cavitation activity, with a wide range of frequencies characteristics of the background turbulent flow and noise. At point d, close to the base of the cylinder, a very high signal with frequency significantly lower than the vortex-shedding frequency can be observed. This corresponds to the cavities moving very slowly within the low-pressure recirculating region, similar to the conclusion drawn from Fig. 7. As we move from d to f (Fig. 15), a distinct frequency can only be identified at f, which lies in the extended shear layer. Beyond the location where the shear layer intersects the centreline (corresponding to point f), there is a dominant frequency at twice the vortex-shedding frequency, representing the vortex shed from either side of the cylinder. The doubling of frequency in the merged shear layer can be utilized in finding the formation length (Lf ), i.e. the distance from the centre of the cylinder to the shear-layer crossover point. Fig. 17 shows that Lf increases with a reduction in σ /σ i , which may be a consequence of the growth in cavity size in the wake. This organization of cavitating structure in the subcritical flow regime seen here is consistent with the flow structure observed by Maekawa and Mizuno (1967) at a similar Re. To compare the frequency-domain information obtained from the high-speed images with other independent measurements, dynamic pressure measurements were performed at the rearward base of the cylinder. Pressure fluctuations arise due to cavitating and non-cavitating flow conditions. In order to bring out the influence of cavitation on pressure fluctuation over and above the non-cavitating flow, we have normalized the two signals as normalized amplitude (20log P Pcav (ω(ω) ) ). The data at different frequennoncav cies are plotted in Fig. 18. The dominant frequency at lower cavitation number (σ /σi = 0.50) here is the shedding frequency at 253 Hz, which matches closely with that obtained from the highspeed images. This shows that the frequency information from cavitating structures corroborates well with independent measurements of dynamic pressure fluctuations. It can be also noted from Fig. 18 that there is a reduction in the shedding frequency with a reduction in cavitation number. This can be explained in terms of the formation length. As pointed out by Gerrard (1966) and Gnanaskandan and Mahesh (2016), the increase in formation length is accompanied with a reduction in the shedding frequency. With the lowering of cavitation number the formation length increases (Fig. 17) and so the vortex shedding frequency reduces. 5. Conclusions This paper has presented results on cavitation in the near wake of a cylinder. To the best of our knowledge, this is the first detailed analysis of cavitating flow structures in the shear layer and the wake of a cylinder. We have established that at the subcritical Reynolds number, cavities originate primarily in the free shear
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Fig. 16. Frequency spectrum of time series of greyscale level g(x, y, n) at different points as shown in Fig. 15 for σ /σi = 0.50: (a ) x/d = 1, y/d = 0.9; (b) x/d = 1, y/d = 0.45; (c ) x/d = 1, y/d = 0; (d ) x/d = 0.6, y/d = 0; (e ) x/d = 1.75, y/d = 0; ( f ) x/d = 2.5, y/d = 0.
layer but in neither the wake nor the attached boundary layer of the cylinder. Spanwise variation of cavitation activity is evident, with a low-activity region surrounded by high-cavitationfluctuation regions on either side in the spanwise direction. The correlation length decreases from about 3.5 at higher cavitation numbers to about 1 at lower cavitation numbers. On the basis of the magnitude of the frequency signature, it is possible to identify three different regimes with distinct frequency content: in the shear layer, the vortex-shedding frequency is observed, while broadband signals are noted in the wake and the outer layer. These three zones resemble the zones defined by Maekawa and Mizuno (1967) on the basis of velocity fluctuations measured in a noncavitating flow. The frequencies from the measured pressure fluctuations on the rearward side of the cylinder correlate well with the frequencies obtained from images. Acknowledgement The authors acknowledge the partial financial support of this work by the Naval Research Board of India.
Fig. 17. Variation of formation length (Lf ) with cavitation number.
Fig. 18. Frequency spectrum of pressure fluctuations on the cylinder surface for different cavitation numbers.
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References Aeschlimann, V., Prothin, S., Barre, S., Djeridi, H., 2012. High speed visualizations of the cavitating vortices of 2d mixing layer. Eur. J. Mech. B/Fluids 31, 171–180. Arakeri, V.H., Shanmuganathan, V., 1985. On the evidence for the bubble interference on cavitation noise. J. Fluid Mech. 159, 131–150. Bilus, I., Bombek, G., Hocevar, M., Sirok, B., Cencic, T., Petkovsek, M., 2014. The experimental analysis of cavitating structure fluctuations and pressure pulsations in the cavitation station. J. Mech. Eng. 60(3), 147–157. Bloor, M.S., 1964. The transition to turbulence in the wake of a circular cylinder. J. Fluid Mech. 19, 290–304. Bloor, M.S., Gerrard, J.H., 1966. Measurements on turbulent vortices in a cylinder wake. Proc. Royal. Soc. Lond. A 319–342. Dular, M., Bachert, B., Stoffel, B., Sirok, B., 2004. Relationship between cavitation structures and cavitation damage. Wear 257, 1176–1184. Fry, S.A., 1984. Investigating cavity/wake dynamics for a circular cylinder by measuring noise spectra. J. Fluid Mech. 42, 187–200. Gerrard, J.H., 1966. Mechanics of the formation region of vortices behind bluff bodies. J . Fluid Mech. 25, 401–413. Gnanaskandan, A., Mahesh, K., 2016. Numerical investigation of near-wake characteristics of cavitating flow over a circular cylinder. J. Fluid Mech. 790, 453–491. Griffin, O.M., Votaw, C.W., 1972. The vortex street in the wake of a vibrating cylinder. J. Fluid Mech. 51, 31–48. Hinsberg, N.P.H., 2015. The reynolds number dependency of the steady and unsteady loading on a slightly rough circular cylinder :from subcritical up to high transcritical flow state. J. Fluids Struc. 55, 526–539. Kiya, M., Matsumura, M., 1988. Incoherent turbulence structure in the near wake of a normal plate. J. Fluid Mech. 190, 343–356. Kumar, P., Chatterjee, D., Bakshi, S., 2016. Experimental investigation of cavitation behind a circular cylinder in cross-flow. In: 9th International Conference on Thermal Engineering: Theory and Applications, March 24–26, Abu Dhabi, UAE. Kumar, T.M.P., 2012. Experimental and Numerical Investigation of Non-Cavitating and Cavitating Flows over S-shaped Hydrofoil. Ph.D. thesis. Indian Institute of Technology, Madras, India. Kumar, T.M.P., Kumar, P., Chatterjee, D., 2014. Cavitation characteristics of s-blade used in fully reversible pump-turbine. ASME J. Fluids Eng. 136(5), 1–15. Laberteaux, K.R., Ceccio, S.L., Mastrocola, V.J., Lowrance, J.L., 1998. High speed digital imaging of cavitating vortices. Expt. Fluids. 24, 489–498. Lehmkuhl, O., Rodrguez, I., Borrell, R., Chiva, J., Oliva, A., 2013. Low-frequency unsteadiness in the vortex formation region of a circular cylinder. Phys. Fluids 25, 085109. Linke, W., 1931. Neue messungen zur aerodynamik des zylinders. insbesondere seines reinen Reibungswiderstandes. Phys. Zr. 32, 900. Maekawa, T., Mizuno, S., 1967. Flow around the separation point and in the nearwake of a circular cylinder. Phys. Fluids 10, S184–S186. Matsudaira, Y., Gomi, Y., Oba, R., 1992. Characteristics of bubble-collapse pressures in a karman-vortex cavity. JSME INT. J., Ser. II. 35, 179–185. Medwin, H., 1977. Counting bubbles acoustically: a review. Ultrasonics. 15, 7–13. Miau, J., Wang, J.T., 1999. Characteristics of low-frequency variations embedded in vortex-shedding process. J. Fluids and Struc. 13, 339–359. Moffat, R.J., 1988. Describing the uncertainties in experimental results. Expt. Thermal Fluid Sci. 3–17. Nagaya, Y., Murase, M., Mizuyama, S., 2011. Detection of cavitation with microphone placed outside piping. J. of Environ. Eng. 6, 426–436.
217
Norberg, C., 1994. An experimental investigation of the flow around a circular cylinder: influence of aspect ratio. J. Fluid Mech. 258, 287–316. Perrin, R., Braza, M., Cid, E., Cazin, S., Moradei, F., Barthet, A., Sevrain, A., Hoarau, Y., 2006. Near-wake turbulence properties in the high reynolds number incompressible flow around a circular cylinder measured by two and three-component piv. Flow Turbulence Combust 77, 185–204. Ramamurthy, A., Bhaskaran, P., 1977. Constrained flow past cavitating bluff bodies. ASME J. Fluids Eng. 99, 717–726. Rao, B.C.S., Chandrasekhara, D.V., 1976. Some characteristics of cavity flow past cylindrical inducers in a venturi. ASME J. Fluids Eng. 98, 461–468.. Reisman, G.E., Wang, Y.C., Brennen, C.E., 1998. Observations of shock waves in cloud cavitation. J. Fluid Mech. 355, 255–283.. Roshko, A., 1954. On the development of turbulent wakes from vortex streets. NACA Rep. 1191. Roshko, A., 1961. Experiments on the flow past a circular cylinder at very high reynolds number. J. Fluid Mech. 10, 345–356. Roshko, A., 1993. Perspectives on bluff body aerodynamics. J. Wind Eng. Indust. Aerodyn. 49, 79. Saito, Y.A., Sato, K., 2003. Cavitation bubble collapse and impact in the wake of circular cylinder. Fifth International Symposium on Cavitation (cav2003), Osaka, Japan. Sarkar, K., Shi, W.T., Chatterjee, D., Forsberg, F., 2005. Characterization of ultrasound contrast microbubbles using in vitro experiments and viscous and viscoelastic interface models for encapsulation.. J. Acoust. Soc. Am. 118, 539–550. Sato, K., Taguchi, Y., Hayashi, S., 2013. High speed observation of periodic cavity behavior in a convergent-divergent nozzle for cavitating water jet. J. of Flow Control, Meas. Visual. 1, 102–107. Schewe, G., 1983. On the force fluctuations acting on a circular cylinder in cross-flow from subcritical up to transcritical reynolds numbers. J Fluid Mech. 133, 265–285. Shalnev, K., 1965. Boundary effect on cavitating flow past a cylinder. J. Appl. Mech. Tech. Phys. 3, 103–108. Szepessy, S., 1994. On the spanwise correlation of vortex shedding from a circular cylinder at high subcritical reynolds number. Phys. Fluids 6, 2406–2416. Szepessy, S., Bearman, P., 1992. Aspect ratio and end plate effects on vortex shedding from a circular cylinder. J. Fluid Mech. 234, 191–217. Unal, M.F., Rockwell, D., 1988. On the vortex formation from a cylinder part 1. the initial instability. J. Fluid Mech. 190, 491–512. Varga, J., Sebestyen, G., 1966. Experimental investigation of cavitation noise. Houille Blanche 8, 905–910. Wei, T., Smith, C.R., 1986. Secondary vortices in the wake of circular cylinders. J. Fluid Mech. 169, 513–533. Weidman, P.D., 1968. Wake Transition and Blockage Effects on Cylinder Base Pressures. Ph.D. thesis. California Institute of Technology, Pasadena, California. West, G.S., Apelt, C.J., 1982. The effects of tunnel blockage and aspect ratio on the mean flow past a circular cylinder with reynolds numbers between 104 and 105 . J. Fluid Mech. 114, 361–377. Williamson, C.H.K., 1996. Three-dimensional wake transition. J. Fluid Mech. 328, 345–407. Zdravkovich, M.M., 1997. Flow around Circular Cylinders. Fundamentals. Oxford University Press, New York.