Wavelet analysis techniques in cavitating flows

Abstract

Cavitating and bubbly flows involve a host of physical phenomena and processes ranging from nucleation, surface and interfacial effects, mass transfer via diffusion and phase change to macroscopic flow physics involving bubble dynamics, turbulent flow interactions and two-phase compressible effects. The complex physics that result from these phenomena and their interactions make for flows that are difficult to investigate and analyse. From an experimental perspective, evolving sensing technology and data processing provide opportunities for gaining new insight and understanding of these complex flows, and the continuous wavelet transform (CWT) is a powerful tool to aid in their elucidation. Five case studies are presented involving many of these phenomena in which the CWT was key to data analysis and interpretation. A diverse set of experiments are presented involving a range of physical and temporal scales and experimental techniques. Bubble turbulent break-up is investigated using hydroacoustics, bubble dynamics and high-speed imaging; microbubbles are sized using light scattering and ultrasonic sensing, and large-scale coherent shedding driven by various mechanisms are analysed using simultaneous high-speed imaging and physical measurement techniques. The experimental set-up, aspect of cavitation being addressed, how the wavelets were applied, their advantages over other techniques and key findings are presented for each case study.This paper is part of the theme issue 'Redundancy rules: the continuous wavelet transform comes of age'.

Keywords: bubble dynamics; cavitation; nucleation; wavelets.

Figure 1.

Schematic of the variable pressure water tunnel at AMC showing circuit architecture for continuous removal of microbubbles or large volumes of injected incondensable gas and ancillaries for microbubble seeding and for degassing of water. Microbubbles may be either injected for modelling cavitation nucleation or generated by the cavitation itself. All dimensions are in metres. Lower left: photograph of a polydisperse plume created by a microbubble injector. The plume has been measured with shadowgraphy and the histogram in terms of bubble production is in the lower right.

Figure 2.

The intensity of scattered light as a function of the viewing angle for a 30 μm bubble in water illuminated by perpendicularly polarized 532 nm light.

Figure 3.

Use of wavelets to analyse fringe patterns obtained by interferometric Mie Imaging of microbubbles. A schematic of the optical arrangement used (a) indicates how the raw image (b) was generated. The image shows the presence of two bubbles of different sizes, evidenced by the differing number of fringes. (c) Sum of pixel intensities down the image. (d) Magnitude of the Morlet wavelet transform across this signal (|T(x, λ)|), correctly identifying the fringe spacing (λ) for the two different sized bubbles.

Figure 4.

Time-series (a) from the CSM showing acoustic excitation events due to microbubble explosion, growth and collapse. (b) The output of the filtering technique, showing the smoothing that occurs between consecutive events. (c) The CWT scalogram (absolute value using the Morlet wavelet). (d) Time series of the mean wavelet power between f = 2¹⁴ Hz and f = 2¹⁸ Hz.

Figure 5.

Cumulative background nuclei distribution in the AMC cavitation tunnel [51]. Nuclei concentration (C) is plotted against the water tension (p_c − p_v), where p_c is the critical pressure at which the nuclei are activated in the venturi throat. The secondary horizontal axis is the bubble size (d₀) in the tunnel test section associated with the critical pressure value.

Figure 6.

Schematic side views experimental set-up for investigating break-up of millimetre bubbles in a turbulent shear layer. A plane jet is created between confining side walls in a surrounding water volume below which a bubble train is released. The bubble break-up is imaging using high-speed shadowgraphy and acoustic emissions recorded simultaneously with a hydrophone.

Figure 7.

A sequence of photographs showing a single 3 mm bubble breaking up into three smaller bubbles with turbulence. Firstly, (a, b) the bubble is stretched and deformed by the turbulence, (c) before releasing a 0.2 mm bubble (highlighted in blue). (d) The main bubble later splits into two bubbles (1.9 mm and 2.6 mm). The timing of the images is indicated by the grey vertical lines in figure 8.

Figure 8.

Real value of the Morlet wavelet transform of a hydrophone signal capturing the break-up of a bubble as it encounters a turbulent shear layer. (b–c) Show the decaying oscillations of the bubble constituents, as slices through the wavelet data, for three different frequencies. These frequencies are proportional to the bubble size [60].

Figure 9.

Photographs of cavitation about a foil for two different seeding conditions. (a) No additional nuclei are supplied in the freestream, while on (b) there is an abundance of weak nuclei that are continually activated.

Figure 10.

(a–d) Absolute value of the Morlet wavelet transform on the lift signal for the cavitating foil without (a) and with (b) additional seeding. The time-average of the transforms are given in (c) showing the frequency shift with the addition of seeding. The FTs are given in (d).

Figure 11.

Wavelet analysis of a cavitating hydrofoil (nuclei deplete flow). (a) The time history of the lift signal. (b) The real value of the Morlet wavelet transform. (c,d) Space–time plots at and , respectively, showing the cavity extent as it varies in time.

Figure 12.

(a) Time history of the lift signal. (b) The real value of the Morlet wavelet transform. (c,d) Space–time plots at and , respectively, showing the cavity extent as it varies in time.

Figure 13.

Cross-sectional plan view of the sphere assembly located on the streamwise axis of the water tunnel test section. Surface dynamic pressure and high-speed photography were measured simultaneously to investigate spectral content and shedding modes of the cloud cavitation. High-speed and still photography were taken horizontally from the side as indicated.

Figure 14.

Side-on photographs of the cavitation about a sphere at σ = 0.80. (a) Image shows asymmetric cavity shedding, while the structures are axisymmetric in (b).

Figure 15.

PSD of sphere surface pressure at σ = 0.80 derived using the FT (Welch) and the CWT using a Morlet wavelet with several non-dimensional frequencies. The FT identifies three peaks, while the third peak is only resolved with the CWT if the non-dimensional frequency is greater than the typical value of 6.

Figure 16.

Real value of the CWT for the near (a) and far (b) pressure taps. The three horizontal lines are the three peak frequencies from f₁ to f₃ ascending. The vertical lines are guides to highlight events that are out of phase (f₁ at the left-hand vertical line), or in-phase (f₂ at the right-hand vertical line).

Figure 17.

Sample time series of the pressure coefficient on the surface of a cavitating sphere. These time series were selected to exemplify when each of the three different frequencies, f₁ (a), f₂ (b) and f₃ (c) were dominant. The filled contours in rows two and three are the real values of the Morlet and DOG2 wavelet transforms, respectively, and the horizontal lines indicate the frequencies of interest. The spatio-temporal map of cloud cavity position is given in the bottom row, where the horizontal line indicates the downstream location of the pressure tap.

Figure 18.

Time series of the pressure coefficient on the surface of a cavitating sphere for σ = 0.75 where only two modes are present. The two pressure taps are located on opposite sides of the sphere. The cross-wavelet transform, here using the DOG2 wavelet, allows for an easy comparison of the phasing of different temporal events, and the coherence between the two signals. The XSD indicates that the first frequency is out-of-phase and the second frequency to be in-phase.

Figure 19.

Pressure coefficient on the surface of a cavitating sphere for three selected time series corresponding to each of the three frequencies. The two pressure taps are located on opposite sides of the sphere. The absolute value of the Morlet cross-wavelet transform is given by the filled contours representing the coherence between the two signals at that Strouhal number. Coherence is evident for only f₁ and f₂. The overlaid arrows represent the phase difference between the two signals (in-phase signals are indicated by an arrow to the right, a π/2 phase difference is indicated by an arrow upwards). The right-most figure is a time-integral of the real part of the wavelet transform, which indicates in a broad sense whether the two signals are in phase (positive) or out-of-phase (negative).