2.1. Test facility and flow conditions

The experiments were conducted in the cavitation tunnel at the Institute of Fluid Engineering, Zhejiang University. The cross-section of the tunnel is $200.5 \times 200.5$ mm$^2$ with a total length of 1020 mm. The contraction of the nozzle is 9, resulting in a maximum velocity of 10 m s$^{-1}$. The turbulent intensity is controlled below 0.5 %. The tip vortex cavitation was produced by an NACA $66_2$-415 elliptic hydrofoil, mounted along the symmetry plane of the sidewall of the test section, as shown in figure 1. The maximum chord ($c$) and span ($s$) of the model were 83.7 mm and 100.1 mm, respectively. The free-stream velocity $u_{\infty }$ was set to 5.0 m s$^{-1}$ for all the test cases, giving a chord-based Reynolds number of $Re_{c}=u_{\infty }c/\nu =4.3\times 10^{5}$, where $\nu$ is the kinematic viscosity of the fluid. The cavitation number is defined as $\sigma =(\,p_{\infty }-p_{v})/0.5\rho u_{\infty }^{2}$, where $p_{\infty }$ is the static pressure at the central location of the water tunnel, $p_{v}$ is the vapour pressure which is determined by the water temperature and $\rho$ is the water density. For the $p_{\infty }$ of 100 kPa and $p_{v}$ of 2.6 kPa at a temperature of 294 K, the cavitation number $\sigma$ is 7.8. The angle of attack was set to $9^{\circ }$ to anticipate vortex cavitation. The $x$, $y$ and $z$ axes correspond to the streamwise, wall-normal and spanwise directions, respectively. The origin of the coordinate system ($o$) is located at the hydrofoil tip.

Figure 1.

Conceptual sketch of tomographic PIV set-up: (a) laser illumination and imaging system, (b) fields of view (FOVs).

The tip vortex cavity is visualized by a high-speed camera (Phantom VEO-340L, $2560\times 1600$ pixels) from the suction side of the hydrofoil, illuminated by a backlight LED. The camera is equipped with objectives of 105 mm focal length. The camera sensor was cropped into $1920\times 1080$ pixels, resulting in an acquisition frequency of 1.5 kHz. The cavity boundary is extracted based on the light intensity gradient, obtained by the Canny edge detection method (Canny Reference Canny1986). The cavity diameter is determined by the wall-normal distance between the top and bottom boundaries of the cavity at each streamwise location.

2.2. Tomographic PIV

The tomographic PIV system features four high-speed cameras (two Photron FASTCAM Mini AX100 and two Photron FASTCAM SA4, $1024\times 1024$ pixels) arranged in a cross-like configuration with a maximum aperture angle of $40^{\circ }$, as shown in figure 1(a). The cameras were equipped with objectives of 105 mm focal length, which were tilted to match the Scheimpflug condition. The aperture was set at $f_{\#} = 22$. In order to eliminate the influence of laser reflection caused by the cavity interface, the flow is seeded with fluorescent particles with a density of 1188 kg m$^{-3}$ (Rhodamine B dye, $35\,\mathrm {\mu }{\rm m}$ average diameter). The particle concentration is 0.02 ppp to avoid the opaqueness effect. The tracer particles were illuminated by an Nd:YLF high-speed laser (Vlite-Hi-527–50, 50 mJ per pulse at 1 kHz). The fluorescent dye emits red light of 584 nm wavelength when illuminated by the 527 nm laser. The camera lens was equipped with long-pass filters, eliminating light with wavelengths below 560 nm. As a result, only the light emitted by the fluorescent particles was received. The laser sheet was expanded to 15 mm in the spanwise direction to capture the tip vortex. A mirror was set above the tunnel to reflect the laser sheet back, increasing the illumination intensity above the hydrofoil. The pulse separation time was set to $100\,\mathrm {\mu }{\rm s}$, yielding a particle displacement of approximately 10 pixels in the free stream for all flow conditions. In order to capture the development of tip vortex cavitation, two domains of interest were measured. The fields of view were both $53.4(x)\times 28.6(y)\times 15(z)$ mm$^{3}$. The streamwise range covers from upstream of the hydrofoil leading edge until 0.9 times the hydrofoil chord in the wake ($x/c = [-0.3, 0.9]$), as shown in figure 1(b). The camera sensors were cropped in the vertical direction for both measurement domains, yielding an active size of $1024\times 512$ pixels. The resultant digital image resolution is 24.6 pixels mm$^{-1}$. The ensemble size of the dataset is 500 snapshots, acquired at 3200 Hz.

The system synchronization, image recording and processing were performed using LaVision Davis 10. The relation between object and image space is obtained by three-dimensional physical calibration using a dual-plane (Type 585) calibration target. Volume self-calibration was performed to further reduce the calibration disparity to less than 0.1 pixels (Wieneke Reference Wieneke2008). The camera simultaneous multiplicative reconstruction technique was used for volume reconstruction (Atkinson & Soria Reference Atkinson and Soria2009). Within the cavity, whose diameter is around 75 % of the vortex diameter, the particle concentration becomes relatively sparse, yielding 4–6 particles. The motion tracking enhancement algorithm in time-marching mode (Novara, Batenburg & Scarano Reference Novara, Batenburg and Scarano2010; Lynch & Scarano Reference Lynch and Scarano2015) was applied to further reduce the ghost particles and improve the reconstruction quality for low particle concentration conditions using several successive exposures. The velocity field is initially calculated, providing estimation for the following snapshots. The volume correlation has a final window size of $32\times 32\times 32$ voxels ($1.48\times 1.48\times 1.48$ mm$^{3}$) with 75 % overlap, resulting in a vector pitch of 0.37 mm. With the tip vortex diameter of approximately 4 mm in the current experiment, more than 10 vectors are obtained along the radial direction of the vortex, enabling the detection of the tip vortex system. The final uncertainty of the velocity field of each condition is summarized in table 1, which is estimated following Ye, Schrijer & Scarano (Reference Ye, Schrijer and Scarano2016) and Sciacchitano & Wieneke (Reference Sciacchitano and Wieneke2016).

Table 1.

Uncertainty parameters of 500 snapshots for three flow conditions, where $\varepsilon _{u}$ represents the uncertainty of the mean velocity, $\varepsilon _{(u')}$ is the uncertainty of the root mean square (r.m.s.) of the velocity fluctuations and $U_{\omega _{x} }$ represents the random errors of the streamwise vorticity.

2.3. Proper orthogonal decomposition

Proper orthogonal decomposition is an effective reduction technique for flow analysis (Berkooz, Holmes & Lumley Reference Berkooz, Holmes and Lumley1993), which decomposes the velocity fluctuation fields into multiple spatially correlated orthogonal modes, representing the spatial distribution of the most energetic fluctuation component (Van Oudheusden et al. Reference Van Oudheusden, Scarano, Van Hinsberg and Watt2005). In this work, the snapshot POD method (Sirovich Reference Sirovich1987) is applied to decompose the streamwise vorticity fluctuations ($\omega ^\prime$) into orthonormal modes $\varphi _{n}(x,y,z)$ and time-dependent amplitude coefficients $\alpha _{n}(t)$ as

(2.1)\begin{equation} \omega^{\prime}(x, y, z, t)=\sum_{n=1}^{N} \alpha_{n}(t) \varphi_{n}(x, y, z), \end{equation}

where $N$ represents the number of snapshots. Each instantaneous vorticity fluctuation obtained by PIV is aligned to one column of a data matrix X=$[\omega ^\prime (t_1), \omega ^\prime (t_2),\ldots, \omega ^\prime (t_N)]$. And each mode can be written as a combination of snapshots:

(2.2)\begin{equation} \varphi_{n}(x, y, z)=\frac{1}{N \lambda_{n}} \sum_{i=1}^{N} \alpha_{n}\left(t_{i}\right) \omega^{\prime}\left(x, y, z, t_{i}\right). \end{equation}

Here, $\lambda _{n}$ values represent the energy contributions of POD modes and are automatically sorted in descending order.

5.1. Unsteady vortex structures

The emergence of vortex cavitation leads to unsteady flow features in and around the tip vortex. The subscale vortical structures contribute to the production of velocity and pressure fluctuation energy. The instantaneous flow organization of vortex structures both upstream and downstream is visualized using the $\lambda _{2}$ criterion (Jeong & Hussain Reference Jeong and Hussain1995) and colour coded by the non-dimensional streamwise velocity $u/u_{\infty }$ for three flow conditions in figure 10–12. The tip vortex originating from the hydrofoil leading edge remains the dominant flow feature for all conditions. According to the instability analysis of the streamwise vortex, Kelvin waves propagate over the surface of the vortex due to the strain effect, leading to the vortex internal displacement and deformation (Moore & Saffman Reference Moore and Saffman1975; Leweke, Le Dizes & Williamson Reference Leweke, Le Dizes and Williamson2016).

Figure 10.

Instantaneous flow organization for both upstream (left) and downstream (right) domains of wetted flow, detected by the $\lambda _{2}$ criterion and colour coded by the non-dimensional axial velocity $u/u_{\infty }$: (a) side view, (b) top view and (c) zoomed side view.

Figure 11.

Instantaneous flow organization of breathing mode cavitation: (a) side view, (b) top view and (c) zoomed side view.

Figure 12.

Instantaneous flow organization of double-helical cavitation: (a) side view, (b) top view and (c) zoomed side view.

For the wetted flow case in figure 10, wavy motion develops over the cylindrical-shaped tip vortex surface (figure 10c), similar to the observation of previous numerical simulations (Asnaghi et al. Reference Asnaghi, Svennberg and Bensow2020a; Cheng et al. Reference Cheng, Long, Ji, Peng and Farhat2021; Xie et al. Reference Xie, Liu, Jiang and Huang2021). Vortex shedding appears over the suction side of the hydrofoil due to flow separation (figure 10(a,b) side and top view). The former structures move towards the hydrofoil tip and interact with the tip vortex around $x/c = 0.1$, leading to noticeable streamwise undulation and diameter variation of the streamwise vortex. The time sequences of the unsteady motion are provided in supplementary movies 3 and 4. The secondary vortices shed from the trailing edge with discontinuous character and decay rapidly downstream. From the top view, periodic shedding of spanwise substructures forms due to flow separation in the hydrofoil wake. The former substructures originate from the secondary vortex filaments, being entrained into the tip vortex in the range of $x/c$ = [0.2, 0.4] and leading to local stretching of the tip vortex structure. (see figure 10(b), top right and supplementary movies 3 and 4).

Under cavitating conditions, the complexity and unsteadiness of the three-dimensional structure increase noticeably with the decrease of cavitation number (figures 11 and 12), indicating the contribution of cavity surface fluctuations on the unsteadiness of the vortical structures. Stronger vortex shedding is produced over the hydrofoil with decreasing cavitation number (top view), leading to more intense interaction and vortex stretching. Arndt et al. (Reference Arndt, Pennings, Bosschers and Van Terwisga2015) claimed that the waves over the cavity surface travel upstream and interact with the boundary layer near the tip, causing violent flow instabilities (Arndt et al. Reference Arndt, Pennings, Bosschers and Van Terwisga2015). For breathing mode cavitation (figure 11a,b), the tip vortex exhibits strong vortex deformation at the interaction point. Hairpin-like structures emerge over the tip vortex due to the intensified shear layer instability around the cavity interface, featuring an arch shape (Stinebring et al. Reference Stinebring, Farrell and Billet1991; Asnaghi et al. Reference Asnaghi, Svennberg and Bensow2020a). These structures appear when the tip vortex detaches from the hydrofoil ($x/c > 0.1$). The latter vortex bifurcates into two branches at $x/c = 0.2$ and 0.4, which quickly reconnect into a single vortex (figure 11c). The legs of the hairpin vortex align in the streamwise direction with a distance of 0.04$c$. Different from the cylinder-like structures in wetted flow, the tip vortex in breathing mode cavitation is closer to a prismatic shape with an apparent spiral around the vortex centreline. The notable temporal shape and diameter variations of the tip vortex are produced (see supplementary movies 5 and 6). However, its periodic character is not as clear as the volume variation of the cavity shown in figure 2(b).

For the double-helical cavitation (figure 12a,b and supplementary movies 7 and 8), the separated vortex structures become more populated and rather dominant. The intensified interaction between flow separation and tip vortex contributes to more prominent streamwise undulation and deformation of the streamwise vortical structure. The spatial periodic shape variation of the twisted cavity leads to vortex stretching (Ji et al. Reference Ji, Luo, Arndt and Wu2014). The notable growth of Kelvin–Helmholtz instability over the tip vortex produces highly populated hairpin vortices with increased size compared with that of the breathing mode cavitation. The former vortex grows into a complete hairpin shape. The legs of the hairpin ride over the tip vortex, yielding a size of 0.05$c$ (figure 12c). Both the fluctuations of the cavity surface and the interaction between the tip vortex and the separated vortices should be responsible for this phenomenon. The deformation of the tip vortex implies faster growth of elliptical instability in the cavitation flow (Kerswell Reference Kerswell2002). Although the twisted shape corresponding to that of the cavity (figure 2a) can still be identified, the smaller-scale features prevail in the unsteady flow field, revealing the effect of the elliptical-shaped cavity on perturbation growth.

5.2. Tip vortex wandering

The development of an unsteady tip vortex with streamwise undulation leads to displacement of the vortex centre, corresponding to the wandering motion. The vortex centre is calculated by the (4.1a,b). The spatial distributions of the vortex centres of 500 snapshots at three different streamwise locations $(x/c=0.4, 0.6, 0.8)$ are shown in figure 13. The distribution of vortex centres falls in the range of elliptical shapes for all streamwise locations. Under the wetted flow condition, the vortex centre distribution of the selected three streamwise cross-sections ($x/c=0.4, 0.6, 0.8$) appears to be a round shape, as shown in figure 13(a). The lengths of the long and short axes of the 90 % confidence ellipses are both 0.007$c$.

Figure 13.

The vortex centre distribution at different streamwise locations with 90 % confidence ellipses: (a) wetted flow, (b) breathing mode cavitation and (c) double-helical cavitation. The red star is the centre of each ellipse, denoting the mean vortex centre.

For cavitating conditions in figure 13(b,c), the vortex centre becomes more dispersedly distributed, revealing a stronger wandering motion (Edstrand et al. Reference Edstrand, Davis, Schmid, Taira and Cattafesta2016; Qiu et al. Reference Qiu, Cheng, Xu, Xiang and Liu2021). The cavity volume and shape variation leads to an elliptical distribution of the vortex centres. For the breathing mode cavitation (figure 13b) the long axis of the ellipse aligns at an angle of $7^{\circ }$ and $20^{\circ }$ with respect to the horizontal axis at $x/c = 0.4$ and 0.6, respectively, indicating spanwise wandering motion. For double-helical cavitation, the angle increases to approximately $80^{\circ }$. The long axis aligns closer to the wall-normal direction, indicating a significant change in the dominant wandering direction (figure 13c). The mean vortex centre moves closer to the root at the selected locations with decreasing cavitation number, revealing a faster initial vortex growth and spanwise development. The trajectory of the tip vortices is varied by cavitation onset in both instantaneous and time-averaged flow fields (§ 4), which is different from the previous assumption of the invariant trajectory (Arndt et al. Reference Arndt, Arakeri and Higuchi1991).

7.1. Patterns of POD modes

7.1.1. Wetted flow condition

For the wetted flow condition, the modes 1, 2 and 15 are selected as the representative POD modes in the upstream domain, as shown in figure 16 by the iso-surface of the streamwise vorticity fluctuation component ($\phi /u_{\infty }$). Cross-sections at the featured location of each mode are extracted for ease of interpretation. Mode 1 with the highest disturbance energy exhibits a pair of vorticity fluctuation patches of alternating sign, corresponding to the displacement mode, as shown in figure 16(a). This mode is responsible for the wandering motion of the vortex centre (Del Pino et al. Reference Del Pino, Lopez-Alonso, Parras and Fernandez-Feria2011). It is similar to that induced by the dipole surface oscillation mode of the cavity (Bosschers Reference Bosschers2008) and the streamwise vortex (Leweke & Williamson Reference Leweke and Williamson1998; Chen, Wang & Gursul Reference Chen, Wang and Gursul2018). The vortex filament changes the vorticity distribution on both sides of the vortex core, leading to serpentine centreline displacement (Fabre et al. Reference Fabre, Sipp and Jacquin2006). Mode 2 demonstrates a similar pattern with a node around $x/c = 0.1$ (figure 16b). For both modes, the centreline displacement appears in the vertical direction when $x/c < 0.08$ due to the blockage of the three-dimensional hydrofoil (figure 16ai,bi). The patches detach from the hydrofoil at $x/c = 0.1$ and rotate to an angle of $45^{\circ }$ when $x/c > 0.2$ (figure 16aii,bii), in accordance with the rotation direction of the vortex. The spanwise displacement motion at this stage (figure 16aii,bii) also agrees with the trajectory of the tip vortex in figure 7.

Figure 16.

Vorticity fluctuation component of POD modes for the wetted flow condition ($\phi /u_{\infty } =2\times 10^{-3}$) in the upstream domain: (a) mode 1, (b) mode 2 and (c) mode 15. Cross-plane contours are superimposed at $x/c=0.08,0.2$.

The first clear appearance of the deformation perturbation is in mode 15 in the upstream domain with much lower disturbance energy. It exhibits a quadrant distribution of positive and negative vorticity fluctuation components shown in figure 16(c), following a similar pattern to the most unstable Kelvin modes with a constant axial flow (Lacaze, Ryan & Le Dizes Reference Lacaze, Ryan and Le Dizes2007). The pattern leads to the surface deformation of the tip vortex (Roy et al. Reference Roy, Leweke, Thompson and Hourigan2011), including both size and shape. The deformation mode initially occurs near the trailing edge of the hydrofoil ($x/c > 0.08$), associated with the stress effect caused by the hydrofoil boundary layer. This mode causes the tip vortex to develop into an elliptical shape.

In the downstream domain, modes 1 and 7, which correspond to the first appearance of the two perturbation types, are selected to show the dominant unsteady vortical patterns, as shown in figure 17. Similar to the upstream domain, the most energetic mode 1 (figure 17a) corresponds to the vortex displacement with a pair of elongated vorticity fluctuation patches of opposite sign, undergoing counterclockwise deflection along the streamwise direction. Consistent with the previous studies of the tip vortex, delta-wing vortex or afterbody vortex (Ma et al. Reference Ma, Wang and Gursul2017; Chen et al. Reference Chen, Wang and Gursul2018; Gursul & Wang Reference Gursul and Wang2018), such modes have a very large wavelength due to the transient growth of the perturbations. The deformation mode makes an earlier appearance in mode 7 for the downstream domain (figure 17b), indicating its increased contribution to the fluctuation energy downstream. Vorticity fluctuation patches evolve downstream following a helical trajectory, as shown in figure 17(b). The energy distribution of POD modes indicates that the displacement mode leads to the most energetic perturbation growth of the tip vortex in the wetted flow.

Figure 17.

Vorticity fluctuation component of POD modes for the wetted flow condition ($\phi /u_{\infty } =1\times 10^{-3}$) in the downstream domain: (a) mode 1, and (b) mode 7. Cross-plane contours are superimposed at $x/c = 0.42, 0.76$.

7.1.2. Breathing mode cavitation

The first three POD modes of breathing mode cavitation in the upstream domain are shown in figure 18. The unsteady vortical structure undergoes periodic pulsations due to the volume variation of the cavity. As a result, the disturbance energy represented by the deformation modes (modes 1–3) significantly increases, revealing a close connection between the tip vortex instability and cavity oscillation. For mode 1 shown in figure 18(a), the vortical fluctuation pattern is constrained by the hydrofoil surface when $x/c < 0.1$, leading to a displacement in the vertical direction (figure 18ai). However, the deformation pattern appears downstream ($x/c > 0.1$) and becomes dominant (figure 18aii). The perturbation patterns of modes 2 and 3 resemble the quadrupole distribution, as shown in figure 18(b,c), indicating the importance of the vortex deformation. The pattern of mode 2 has a phase difference of $90^{\circ }$ compared with mode 1 (see figure 18bi,aii). Their linear combination allows for the vortex deformation in all azimuthal directions (Gentile et al. Reference Gentile, Schrijer, Van Oudheusden and Scarano2016).

Figure 18.

Vorticity fluctuation component of POD modes for the breathing mode cavitation ($\phi /u_{\infty } =2\times 10^{-3}$) in the upstream domain: (a) mode 1, (b) mode 2 and (c) mode 3. Cross-planes are selected at $x/c = 0.08, 0.24$ for mode 1 and $x/c = 0.2$ for modes 2 and 3.

In the downstream domain, the first mode with the highest energy still represents vortex elliptical deformation induced by radial volume variations of the cavity, shown in figure 19(a), similar to that in the upstream domain. The pattern reveals vortex deformation and rotation that coincide with cavity diameter change obtained by the high-speed camera (figure 2b). The patches extend over the entire measure domain, which is much larger compared with that of the wetted flow condition (see mode 7, figure 17b). Different from the upstream domain, the displacement mode of the vortex first appears in mode 3 (see figure 19b), indicating that the bending or wandering motion of the tip vortex intensifies downstream even in the cavitating flow. The vorticity fluctuation patches show very little rotation compared with similar modes for wetted flow conditions in figure 17(a).

Figure 19.

Vorticity fluctuation component of POD modes for the breathing mode cavitation ($\phi /u_{\infty } =1\times 10^{-3}$) in the downstream domain: (a) mode 1 and (b) mode 3. Cross-plane contours are superimposed at $x/c = 0.48, 0.78$.

For the breathing mode condition, the POD modes of high fluctuation energy mainly represent the vortex deformation, indicating its dominant role in the growth of vortex instability.

7.2. Power spectral density of POD modes

In order to analyse the spectral content of the POD modes, the power spectral density of the time coefficient matrix of the selected modes is performed, shedding light on the dominant vortex perturbation frequency and the correspondence with the cavity unsteadiness, as shown in figure 20. For the wetted flow (solid black line) and double-helical cavitation (dash black line) conditions, no dominant peak was found for displacement and deformation modes, indicating the absence of periodic motion. The overall spectra indicate a broadband pattern.

Figure 20.

The power spectral density (PSD) functions of the most energetic modes for the three flow conditions.

For breathing mode cavitation, distinct peak frequencies are observed for the deformation mode 1 in both upstream (red line) and downstream (dash blue line) domains. The periodic deformation yields a frequency of 220 and 237 Hz, respectively, similar to that of the periodic cavity volume variation obtained by the high-speed videos in figure 2(b). In order to investigate the mechanism of the breathing mode cavitation, the natural oscillation frequency of a cylindrical bubble (Choi et al. Reference Choi, Hsiao, Chahine and Ceccio2009) is calculated by

(7.1)\begin{equation} f=\frac{\varGamma }{4{\rm \pi}^{2}r_{c}^{2}\sqrt{\ln(r_{D}/r_{c})}}, \end{equation}

where $\varGamma$ represents the circulation and equals 0.03 m$^{2}$ s$^{-1}$, $r_{c}=1.25$ mm is the mean cavity radius and $r_{D}$ represents the half-width of the test section. The natural frequency is 236 Hz, reaching a good agreement with that of the POD modes and cavity diameter fluctuations. The results reveal that the vortex instability and cavity volume variation are related to the excitement of the cavity resonance (Peng et al. Reference Peng, Wang, Li, Xu and Song2017a).

7.3. Instability mechanisms of cavitating tip vortex

The unsteady flow structures around the tip vortex are introduced by wave motion, the interaction with the separated wake and the cavity surface instability. Under cavitating conditions, the surface fluctuations of the tip vortex are significantly intensified due to the shape and volume variation of the cavity and stronger interaction with the unsteady vortex shedding, producing hairpin vortex structures and notable streamwise vortex undulation. The unsteady vortical structure and related instability mode are greatly affected by cavitation.

From the POD analysis, the patterns of unstable vortex perturbations reveal displacement and deformation modes. For the wetted flow, the displacement mode dominates the growth of vorticity fluctuations, similar to the wing-tip vortex of the airfoil (Del Pino et al. Reference Del Pino, Lopez-Alonso, Parras and Fernandez-Feria2011). The former mode is attributed to the self-induced Biot–Savart effect or external shear stress of the boundary layer and the vortex system, leading to the bending and wandering motion of the vortex with directivity (Garmann & Visbal Reference Garmann and Visbal2017a).

For the breathing mode, the periodic volume variation of the cavity induces strong azimuthal instability, which promotes the growth of the deformation mode. The disturbance energy magnitude of the former mode intensifies beyond the level of the displacement mode. The peak frequency and wavelength of the deformation mode are consistent with the diameter fluctuations of the cavity and the natural oscillation mode of the cylindrical bubble, revealing a direct correlation between cavitation and vortex instability. The change of dominant POD mode suggests that the instability of the tip vortex is significantly affected by cavity volume variation, which can be explained by the oscillation of the cylindrical bubble near the equilibrium radius.