# The influence of nucleation on cloud cavitation about a sphere

Katrina L de Graaf, Bryce W Pearce, Paul A Brandner

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Katrina L de Graaf, Bryce W Pearce, Paul A Brandner. The influence of nucleation on cloud cavitation about a sphere. 16th International Symposium on Transport Phenomena and Dynamics of Rotating Machinery (ISROMAC 2016), Apr 2016, Honolulu, United States. hal-01521585

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# The influence of nucleation on cloud cavitation about a sphere

K. L. de Graaf<fnref n="1" />, B. W. Pearce<fnref n="1" />*, P. A. Brandner<fnref n="1" />

<figure><img src="image_2.png" /></figure>

ISROMAC 2016

International
Symposium on
Transport
Phenomena and
Dynamics of

Rotating Machinery

Hawaii, Honolulu

April 10-15, 2016

**Abstract**

The influence of nucleation on cloud cavitation about a sphere is investigated using simultaneous high-speed photography and high-frequency surface pressure measurements. The 0.15 m diameter PVC sphere was located on the axial centreline of a variable-pressure water tunnel with a 0.6 m square test section and tested at a constant Reynolds number of $1.5×10^6$ and a cavitation number of 0.7. The flush-mounted high-frequency pressure sensor was located within the shedding zone of the cloud cavitation and orientated such that it was in the field of view of the high-speed camera. Pressure measurements and high-speed photography were simultaneously obtained at 7 kHz. The time-series of spatially averaged image pixel-intensity was found to lead the pressure measurement. Wavelet analysis was used to compare both data sets obtained for cavitation about the sphere with and without upstream artificial nuclei seeding. PSDs generated from the wavelet analysis showed two peak frequencies for the cavity shedding, with the peaks being more defined for the nucleated case.

**Keywords**

Bluff body shedding — Cavitation — Wavelets

## INTRODUCTION

The repetitive formation, separation and collapse of sheet cavities is generally described as cloud cavitation. Discrete cavity volumes are periodically detached from the growing cavity due to the formation of a re-entrant jet at the cavity closure. The re-entrant jet penetrates or breaks the growing cavity, detaching a portion that condenses to a bubble cloud with downstream advection [1]. This phenomenon has been widely studied, e.g. [2–8] and recently it has been shown that the formation of condensation shockwaves produce another mechanism for instability of partial cavitation [9, 10].

Cavitation about spheres and axisymmetric headforms has also been the topic of numerous studies. Brennen [11] looked at cavity formation behind axisymmetric headforms (and spheres), including the surface appearance and nature of the separated interfacial boundary layer. Arakeri and Acosta [12] investigated cavitation inception and the role of boundary layer separation and turbulent reattachment. Huang [13, 14] also investigated cavitation inception in regions of boundary layer separation for six axisymmetric headforms. Ceccio and Brennen [10] studied the fluctuations of attached cavities on two axisymmetric bodies. Tassin Leger and Ceccio [15, 16] investigated cavity detachment of axisymmetric cavities. Haipeng et al. [17] used high-speed imaging to obtain data of cavitation about blunt and hemispherical axisymmetric bodies for comparison with numerical results. Hu et al. [18] also used high-speed imaging, as well as PIV to study cavitation flow-field interaction about a blunt, axisymmetric body.

<figure><img src="image_3.png" /><figcaption>**Figure 1.** Schematic of sphere experimental setup showing Polyvinyl chloride (PVC) front section and stainless steel rear section. The rear section has a flush-mounted dynamic pressure sensor located within the shedding zone.</figcaption></figure>

Cloud cavitation about a sphere; however, is not well reported on. The present work extends that of Brandner et al. [9] and de Graaf et al. [20] and investigates cloud cavitation developed about a sphere at a high Reynolds number with and without artificial seeding of nuclei. Both surface pressure measurements and high-speed photography are used to obtain the cavity shedding frequencies.

<footnote n="1">¹Australian Maritime College, University of Tasmania, Australia</footnote>

<footnote>*Corresponding author: Bryce.Pearce@utas.edu.au</footnote>
<figure><img src="image_4.png" /></figure>

**Figure 2.** (Top) The un-seeded case shows the classical cavity detachment associated with laminar boundary layer separation. The cavity closure shows re-entrant jet formation with initial evidence of the jet impinging on the cavity surface initiating shockwave formation (see figure 3). (Bottom) For the seeded case where the cavitation is formed from discrete nuclei activations no clear leading edge or point of detachment develops. Despite the discontinuous nature of the cavity volume re-entrant jets still form as do the shockwaves although these are less evident from still photographs and are more apparent from high-speed movies.
<figure><img src="image_5.png" /><figcaption>Figure 3. A still photograph of cavitation, without seeding, showing the propagation of a condensation shockwave after the impingement of the re-entrant on the surface of the growing cavity (see figure 2 (top)).</figcaption></figure>

# 1. EXPERIMENTAL OVERVIEW

A 0.15 m diameter sphere is mounted on the horizontal centreline of the variable-pressure water tunnel at the Cavitation Research Laboratory at the Australian Maritime College, University of Tasmania. The tunnel has a working section of 0.6 × 0.6 × 2.6 m, and the pressure can be varied from 4 - 400 kPa and the velocity from 2 - 12 m/s. The sphere has a Polyvinyl Chloride (PVC) front section and a stainless steel rear section in which a flush-mounted dynamic pressure sensor (PCB 105C02) is located 60° downstream of the sphere centreline (figure 1). The sensor is then in the shedding zone of the cavitation. Nuclei were injected on the stagnation streamline of the sphere. The microbubble generator used produces 1.5×10⁶ nuclei per second in the size range of 5 - 100 µm, with a dominant size range of 10 - 20 µm [21]. Data was recorded from the pressure sensor for 2 seconds at 7000 Hz using a National Instruments PXIe-4497 card. High-speed photography was also recorded simultaneously, using a LaVision HighSpeedStar8 camera with a Nikkor 50 mm f/1.8 lens, operated using DaVis 8.2.0 software and illuminated with two COB LED lamps. An additional longer data series from the pressure sensor was recorded at 1024 kHz for 200 seconds.

High-resolution images were captured using a Canon EOS 50D camera with a 24-70 mm f/2.8 lens illuminated with triggered stroboscopic lighting (DRELLO 1018/LE4040).

# 2. RESULTS

All data was recorded for a cavitation number of 0.7 and a Reynolds number of 1.5×10⁶. The cavitation number, $\sigma$, is given by

$$ \sigma = \frac{p - p_v}{1/2 \rho U^2} \qquad (1) $$

where $p$ is the static pressure at the working section centreline, $p_v$ is the vapour pressure, $\rho$ is the water density, and $U$ is the working section centreline velocity. The Reynolds number, $Re$ is given by

$$ Re = \frac{UD}{\nu} \qquad (2) $$

where $D$ is the sphere diameter and $\nu$ is the kinematic viscosity of water.

Figure 2 (top) shows cavitation about the sphere both with and without artificial seeding of nuclei. For the unseeded case, cells form in the boundary layer on the sphere body and then merge to form a larger cavity that sheds as it is broken up by the re-entrant jet and ensuing shock waves [9]. In this image, the initial evidence of the jet impinging on the cavity surface can be seen where the slightly glassy cavity becomes more textured. For the seeded case, some freestream nuclei are activated as they enter the low pressure region adjacent to the surface forming large single bubble cavities [22], whilst
<figure><img src="image_6.png" /><figcaption>**Figure 5.** Scattter plot of shifted, normalised intensity vs normalised pressure for $\sigma = 0.7$ without (top) and with (bottom) seeding. The general 45° shape of the plots indicates good correlation, while the flat bottom is due to the image intensity reaching a maximum 255 counts (intensity plot is reversed).</figcaption></figure>

<caption>**Figure 4.** (Top) normalised simultaneously acquired pixel intensity ($I'$) and pressure ($P'$) signals for which the phase difference between the two signals is evident. (Bottom) shifted pixel intensity ($I'_s$) and pressure signal. The delay, found using correlation of the two signals, was 8.7 ms.</caption>

at other locations the nuclei form cavities over the sphere surface, growing in the boundary layer. Downstream the cavities merge to larger vapour structures which then exhibit similar breakup physics, despite the discontinuous nature of the cavity volume, to the unseeded case.

From the high-speed photography, time series of pixel intensities were extracted over the region of the pressure transducer. The pixel intensity was averaged over a 65 × 65 pixel area, as determined in a previous investigation [20]. The size of the sampling region was not found to significantly impact the time series produced. The top plot of figure 4 shows the two simultaneously recorded signals for the unseeded case. Both signals have been normalised by subtracting the mean and dividing by the standard deviation, giving the normalised intensity, $I'$, and the normalised pressure, $P'$. Time, $t$, has been non-dimensionalised by $T' = Ut/D$. The intensity signal has been inverted, such that a ‘darker’ average pixel intensity (i.e. clear water) corresponds to a high pressure and a ‘lighter’ average pixel intensity (i.e. white vapour cloud) corresponds with a low pressure. The signals are slightly out of phase, with the image intensity leading the pressure signal. This was previously attributed to an error in the triggering of the camera software [20]. The error has now been resolved, and as the delay is still evident, the phase difference may be attributed to the temporal variation between the streamwise position of the cavity and the pressure magnitude experienced by the surface sensor due to the flow field around the cavity edge. The delay between the signals can be found using cross-correlation; for the unseeded case, the delay is 8.7 ms and for the seeded case, 9.3 ms.

Figure 5 shows scatter plots of the shifted intensity ($I'_s$) vs pressure. The general 45° shape of the plots indicates good correlation. Pearson's correlation coefficients for the

unseeded and seeded cases are 0.71 and 0.75, respectively. The ‘cut-off’ appearance at the bottom of the plots is due to the image intensity reaching a maximum 255 counts (intensity plot is inverted). In a previous study [20] the scatter plots exhibited a non-linearity, indicating a difference in the maximums and minimums of each of the signals. The better correlation in the present data can be attributed to improved lighting settings.

The wavelet analysis method of Torrence and Compo [23] with a Morlet-6 wavelet has been used to analyse the frequency content of the signals. The intensity and pressure time series, shifted to remove the delay, are shown in figure 6, along with the respective wavelet plots. The frequencies, $f$, are represented as Strouhal numbers ($St$), given by $fD/U$. Figure 7 shows the Power Spectral Densities (PSD) calculated from the wavelet analysis of the intensity and pressure signals. The
<figure><img src="image_7.png" /><figcaption>**Figure 6.** Wavelets plots. (Top) without and (bottom) with seeding. Higher frequencies are evident in the bottom plot due to the response of activated nuclei.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>**Figure 7.** PSD of the intensity and pressure signals and also of a longer pressure signal for comparison. (Top) without and (bottom) with seeding.</figcaption></figure>

more resolved PSD of the longer pressure signal (also calculated using wavelets) is also included for reference. The intensity and the pressure signals contain similar frequency content. Two peak frequencies, as observed in [9, 20], are obtained from the spectral analysis of both the pressure and the pixel-intensity signal. These frequencies have been attributed to axisymmetric and non-axisymmetric cavity shedding events. For these conditions, the peak frequencies are more defined for the seeded case, suggesting more concentrated energy in these cavity shedding modes. The spectral content of the longer pressure signal shows less noise in the lower frequencies, as expected.

### 3. CONCLUSIONS

The spectral content of cloud cavitation about a sphere with and without artificial seeding has been investigated for $\sigma = 0.7$ and $Re = 5 \times 10^6$. In the unseeded case, attached sheet cavities form about the sphere surface before breaking up due to impingement of the re-entrant jet on the cavity surface with ensuing shockwave formation. With artificial seeding, bubble cavitation forms at discrete points and no distinct leading edge is evident. Despite the discontinuous nature of the cavity, similar shedding mechanisms to with the non-seeded case were evident for the bulk cavity.

Surface pressure measurements were compared with spatially averaged pixel intensity time-series extracted from high-

speed imaging. Strong correlation was found between the two signals, although a phase shift was evident. Wavelet analysis was used to generate PSDs of the pressure and pixel intensity signals. Two peak frequencies were found in both signals and the PSDs were similar overall in spectral distribution and magnitude. For the conditions tested, the two peaks were more defined for the case with artificial seeding.

### ACKNOWLEDGMENTS

The authors wish to acknowledge the support of the University of Tasmania and the Defence Science and Technology Organisation.

Wavelet software was developed by C. Torrence and G. Compo and was obtained at URL: http://paos.colorado.edu/research/wavelets/

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