3.1. Mean velocities and turbulence

The 2-D PIV measurements are used to characterize the ensemble averaged and root mean square (r.m.s.) values of the axial and vertical velocity components in the shear layer. Figure 3 summarizes the mean flow structure, including profiles of the boundary layer just upstream of the step (figure 3a), contours of ensemble-averaged streamwise velocity $\overline {{u_x}} $ (figure 3b) and profiles in selected locations (figure 3c). The boundary layer profiles, which resolve the log layer but not the viscous sublayer, confirm that the incoming boundary layer is turbulent. The wall shear velocity ${u_\tau }$ ($\sqrt {{\tau _w}/\rho } $, where τ_w is the wall shear stress) is estimated by least square fitting to the log layer profile (Smits, McKeon & Marusic Reference Smits, McKeon and Marusic2011) and the boundary layer height $\delta $ is determined based on the elevation of 99 % of the peak velocity. As summarized in table 1, the boundary layer thickness does not change significantly with velocity, in contrast to typical naturally developing boundary layers, presumably owing to the effect of tripping and acceleration of the flow upstream of the step. The tripping causes an increase in $\delta $ with increasing velocity (Ling et al. Reference Ling, Katz, Fu and Hultmark2017), and the corresponding decrease in acceleration parameter ($\nu /{U^2}\,\textrm{d}U/\textrm{d}x$) is expected to decrease it. The resulting shear Reynolds numbers, $R{e_\tau } = {u_\tau }\delta /\nu $, where ν is the viscosity of water, are 305, 807, 1504 and 2345 for free-stream velocities of 1.45, 5.3, 10.5 and 16 m s⁻¹, respectively. The reattachment length ${x_r}$ is determined based on the streamwise location where the averaged zero velocity line reaches the bottom wall. Its magnitude varies between 5.3 and 6.2 times the step height, consistent with previously published values, e.g. 6h in Jovic & Driver (Reference Jovic and Driver1995) and 6.5h in Spazzini et al. (Reference Spazzini, Iuso, Onorato, Zurlo and Di Cicca2001). As is evident from figure 3(c), the mean velocity profiles in the shear layer nearly collapse after normalizing the horizontal axis by x_r, the vertical axis by h and the velocity by U, with the results at 1.45 m s⁻¹ slightly deviating from the others.

Figure 3. Mean flow structure: (a) a scaled velocity profile of the turbulent boundary layer upstream of the step, showing the log and outer layers. (b) Contours of the ensemble-average streamwise velocity at U = 16 m s⁻¹. (c) Wall-normal profiles of ensemble average streamwise velocity at three streamwise locations at the indicated different speeds and corresponding Re_τ.

Table 1. The reattachment length of the shear layer and the properties of the separating boundary layer for the indicated speeds.

Figure 4 presents ensemble-averaged statistics of velocity fluctuations (u_i) in the shear layer, including contour plots for U = 16 m s⁻¹ (figure 4a–c) and profiles for all velocities in selected location (figure 4d). For the most part, the normalized profiles collapse in the shear layer and in the recirculation zone under it, but there is some deviation at 1.45 m s⁻¹ in the free stream. The peaks in the normalized Reynolds stress components shift downwards and increase in magnitude as the shear layer expands, with the streamwise component being the highest. In each plane, the highest turbulence level is measured slightly below the elevation of maximum velocity gradients. The maximum streamwise turbulence intensity is measured around x/x_r = 0.94 and y/h = −0.5, with values reaching 20 % of the free-stream velocity, and then starts decaying further downstream. In the free-stream velocity fluctuations (both components) fall in the 1 %–2 % of the free-stream velocity, and the Reynolds shear stress vanishes. In general, the Reynolds stress profiles and magnitudes are consistent with previously published data (Kostas, Soria & Chong Reference Kostas, Soria and Chong2002; Nadge & Govardhan Reference Nadge and Govardhan2014). The scaling of the ensembled flow quantities with free-stream velocity is also consistent with the seminal experiments by Eaton (Reference Eaton1980).

Figure 4. Reynolds stress statistics in the shear layer. Contours of: (a) $\sqrt {{u_x}{u_x}} /U$, (b) $\sqrt {{u_y}{u_y}} /U$ and (c) $\sqrt { - {u_x}{u_y}} /U$ at U = 16 m s⁻¹. (d) Wall-normal profiles at the indicated streamwise locations, and at four different speeds.

3.2. Early cavitation events

The cavitation events appear as one or more elongated structures aligned along curved lines that are largely oriented perpendicularly to the spanwise direction (figure 5a). These observations suggest, consistent with prior observations (Katz & O'Hern Reference Katz and O'Hern1986), that cavitation inception occurs preferentially inside the QSVs. The initial growth from a nucleus of size less than 100 μm to a few mm long structure is shorter than 0.25 ms and the sequence lasts for 1–2 ms until desinence. Figure 5(b) shows the spatial distribution of the void fraction of cavitation events, as defined based on the size and shape thresholding criteria, for the two higher speeds, ‘natural’ seeding and cavitation indices ranging between 0.45 and 0.55, i.e. when cavitation is still intermittent. The most likely site is scattered between 0.5 and 0.7x_r, but the peak seems to move slightly upstream with increasing velocity. Furthermore, the cavitation activity increases with decreasing σ (as expected), and for the same σ, with increasing velocity. The latter trend is consistent with the previous observations by Barbaca et al. (Reference Barbaca, Venning, Russell, Russell, Pearce and Brandner2020) as well as the increase in inception index with Reynolds number (Katz & O'Hern Reference Katz and O'Hern1986). As σ decreases from 0.55 to 0.45 at U = 10.5 m s⁻¹, there is a sevenfold increase in the void fraction of cavitation events. As the velocity is increased from 10.5 to 16 m s⁻¹, at the same σ, the peak void fraction jumps up over four times. Incidentally, accounting for the non-cavitating bubbles, which includes nuclei and remnants of previous cavitation events, would increase the peak void fractions by 201 % and 16 % at 10.5 m s⁻¹ and 16 m s⁻¹, respectively, both for σ = 0.5. The cavitation void fraction increase is not due to differences in size of cavities alone. The count of cavities detected are 1680, 383, 165 at 10.5 m s⁻¹ with increasing σ and 9393, 2840 and 716 at 16 m s⁻¹. Statistics on the length, diameter and aspect ratios of cavitation events are provided in figure 5(c). In general, the dimensions of cavities do not change substantially with U and σ, with the most likely diameters being approximately 0.5 mm, and lengths, 1.5 mm. The aspect ratios, that is, the length relative to the diameter, of the structure peak at around 3 for all cases. However, at 16 m s⁻¹, there are significantly more cavities with aspect ratios exceeding 6 compared with those observed at 10.5 m s⁻¹.

Figure 5. (a) Sample perpendicular images for U = 10.5 m s⁻¹ showing the evolution of a cavitation event starting from the nucleus marked by the dashed circle at t = 0 ms. (b) The spatial distributions of ensemble-averaged void fraction for the indicated speed and cavitation indices. (c) The PDFs of length, diameter and aspect ratio of the cavities.

Cavitation inception often occurs at multiple points along a QSV with different events not necessarily occurring at the same time. Figure 6 is a partial series of sample top-view images recorded at 50 kHz for U = 16 m s⁻¹ and σ = 0.452, demonstrating this phenomenon. The entire process consists of multiple growth and collapse events occurring in multiple places and at different times. In some cases, as they grow, the cavities merge forming larger ones. For example, cavity 2 starts growing after cavity 1. They nearly merge at 0.1 ms, and then cavity 1 collapses, while no. 2 lingers. Additional cavities, numbers 5 and 7, originate from fragments of no. 2. However, the nucleus of numbers 3, 6 and 8 does not seem to originate from residual bubbles of no. 2. Note that we are not showing all the images, only selected samples. Each rapid growth or collapse event occurs in approximately 100 μs, while the whole sequence lasts for 1–2 ms. While events may occur along the same tilted vortex, e.g. up to approximately 0.8 ms, at other times they occur in multiple structures, e.g. at t > 0.9 ms.

Figure 6. Sample snapshot top views of cavitation inception at U = 16 m s⁻¹ at the indicated times. Data are recorded at 50 kHz. The flow is from left to right.

Figure 7(a) provides the joint PDFs of cavity length and diameter at 10.5 and 16 m s⁻¹. In the following discussion we divide the cavities to three groups, similar to those discussed in Barbaca et al. (Reference Barbaca, Venning, Russell, Russell, Pearce and Brandner2020). The first contains ‘small spherical’ cavities referring to those smaller than 0.5 mm with aspect ratio lower than 2.0; the second group includes ‘large spherical’ bubbles, i.e. larger than 0.5 mm and aspect ratio of less than 2.0; and the third has ‘elongated structures’, corresponding to cavities with a larger aspect ratio. For convenience, figure 7(a) also shows two lines representing aspect ratios of 2 and 5. For both speeds, the most likely cavities are nearly spherical with diameter of 0.2 mm, which mostly correspond to early phases of growth. With increasing size, the most likely aspect ratio deviates slightly from 1 at 10.5 m s⁻¹, and exceeds 2 at 16 m s⁻¹ with a substantial fraction falling around and even exceeding 5. The next discussion focuses on the growth rate of bubbles. Figure 7(b) shows several samples of the evolution of the cavity length at 16 m s⁻¹. As is demonstrated, the typical initial growth lasts less than 0.1 ms and, after growing at a slower pace for another 0.1–0.2 ms, the cavities either collapse or plateau in size, followed by a second growth and collapse. Figure 7(c) presents PDFs of the normalized collapse and growth rates along the cavity major axis for both speeds, focusing on events with normalized growth rates |dL/dt|>0.2U, i.e. not including cases where the size does not change significantly. For small spherical bubbles, the normalized growth rates at both speeds are faster than the collapse rates. This trend is reversed for the elongated cavities. There is a difference between speeds in the collapse rates for large spherical bubbles, with those at 16 m s⁻¹ being significantly faster. The elongated structures are also more likely to grow at intermediate rates at 10.5 m s⁻¹ compared with those at 16 m s⁻¹ or the spherical structures at the same speed. However, in general, for each of the three groups, once normalized by U the growth (and collapse) rates at the same σ (~0.5) do not seem to vary substantially with velocity. This trend is consistent with that predicted based on an energy balance for nearly spherical bubbles located in a tip vortex by Arndt & Maines (Reference Arndt and Maines2000). They conclude that the axial growth (or collapse) rate of the cavity is proportional to ${(2({p_C} - {p_V})/\rho )^{0.5}}$, where p_C is the pressure in the vortex core. Their experimentally determined proportionality factor is approximately 2. For a steady tip vortex Choi & Ceccio (Reference Choi and Ceccio2007) show proportionality ratios ranging from zero to 1.5, with the values generally increasing with the aspect ratio at the same cavitation index. As will be shown in the next section, where we show and discuss the PDFs the pressure, the characteristic pressure in the QSVs generally scales with U². Hence, for the same σ and aspect ratio the normalized growth rate should not be dependent on the free-stream velocity, consistent with the present findings.

Figure 7. (a) Joint PDFs of cavity diameters and length at U = 10.5 and 16 m s⁻¹ presented using a logarithmic scale. The black and red lines mark aspect ratios (AR = length/diameter) of 2 and 5, respectively. (b) Sample evolution of cavity lengths with time along with their images at selected instances. (c) The PDFs of the axial collapse and growth rates of cavities for the indicated lengths and aspect ratios.

3.3. Pressure fluctuations

To focus the pressure statistics obtained from the time-resolved tomographic measurements on the QSVs, the structures must be detected and tracked. We have tried various forms of conditional sampling, e.g. thresholding based on λ ₂ (the second eigen value of the sum of squares of the symmetric and anti-symmetric parts of the velocity gradient tensor (Jeong & Hussain Reference Jeong and Hussain1995)), with unsatisfactory level of success. Consequently, a multi-dimensional detection method using k-means clustering (Lloyd Reference Lloyd1982) has been adopted. The clustering is based on quantities derived from the velocity gradients experienced by 95 000 synthetic particles placed in the measured flow field and advected in five consecutive time steps. This pseudo-Lagrangian method has been chosen to insure the spatial and temporal continuity of the detected structures. The particle motions are tracked using a fourth-order Runge–Kutta method, with cubic interpolations for the spatial distribution of velocity. The 3-D velocity gradients, obtained via CCM, are used for calculating the vorticity $\omega$_i, λ ₂ and vortex stretching terms ($\omega$_i∂_iu_j). The QSV axis n is identified as being perpendicular to the direction of spatial gradients of the vector sum of $\omega$_x and $\omega$_y. The 3-D position and the following six variables are recorded for each particle at all five times: (i) spanwise vorticity $\omega$_z, (ii) vorticity perpendicular to spanwise-direction, $\omega$_xy, (iii) λ ₂, (iv) projection of the vortex stretching term along the QSV axis (Φ = $\omega$_i∂_iu_jn_j), (v) projection of the strain rate tensor on the axis of the vortex (n_i∂_iu_jn_j) and (vi) the strain state parameter s* (Lund & Rogers Reference Lund and Rogers1994), defined as $s^\ast = {-} 3\sqrt 6 {s_1}{s_2}{s_3}/{({s_1}^2 + {s_2}^2 + {s_3}^2)^{3/2}}$, where s_i are the eigenvalues of the strain rate. Here, s ₁ > s ₂ > s ₃ (note that s ₁ + s ₂ + s ₃ = 0), i.e. s ₁ is the most extensive component, s ₂ the intermediate one and s ₃ the most compressive strain rate eigenvalue. Both forward and backward time steps (±2dt) are used, with dt corresponding to the delay between exposures. For particles that are advected out of the volume, about 10 % of the total, only unidirectional time steps of 4dt are considered. The resulting 45-dimensional dataset (9 variables at 5 times) is divided into 10 clusters using the correlations-based k-means method. This number of clusters has been selected based on silhouette analysis of the data, which minimizes the in-class variance while maximizing the separation between clusters (Rousseeuw Reference Rousseeuw1987). This selection results in clusters with centres that are well separated from each other, and therefore are not sensitive to the selected threshold levels. Before clustering, the mean of each variable is removed and the quantities are normalized by their variance. The clusters with centres that have λ ₂ lower than the mean, as well as $\omega$_xy and vortex stretching magnitude higher than the mean at all 5 times are chosen as QSV candidates. The detected QSV field is projected back to the grid using cubic interpolation. The grid points that have fewer than half of their neighbours classified as QSVs are re-labelled as not being part of a QSV.

To visualize a sample high-dimensional dataset, t-distributed stochastic neighbour embedding (t-SNE) is employed (van der Maaten & Hinton Reference van der Maaten and Hinton2008). This procedure reduces the distribution of data to two dimensions while preserving the probability of inter-particle similarity in the 45-dimensional space. Sample distributions corresponding to one instance (along with ±2dt) are presented in figure 8(a). They show maps of all the same points colour coded by $\omega$_xy, λ _2, pressure and the points denoted as QSVs (1-QSV, and 0-non QSV) based on the criteria listed above. As can be noted, the regions selected as QSVs are organized in clusters that typically have high $\omega$_xy, low λ ₂ and low pressure. To assess the validity of the detection procedures, results have been compared with identification based on threshold values of λ ₂ and $\omega$_xy. Figure 8(b) compares PDFs of the correlations of the pressure with λ ₂ and $\omega$_xy for the detected QSVs with those obtained by conditional sampling involving two conditions, namely that ${\lambda _2} < - 1.7{(U/h)^2}$ and ${\omega _{xy}} > 3.5U/h$, which correspond to the highest 20 % of the measured values. Conditional correlations between variables f and g for conditions F and G are defined as

(3.1)\begin{equation}R(f,g) = \langle (f{|_F} - \langle\, f{|_F}\rangle )(g{|_G} - \langle g{|_g}\rangle )\rangle /{\varsigma _{f{|_F}}}{\varsigma _{g{|_G}}},\end{equation}

where $\langle \,\rangle $ denote spatial averaging for Eulerian statistics and averaging over instances of particles for Lagrangian statistics and ς denote standard deviations. For both correlations, the results of k-means-based detection have higher tails, implying that more events with high correlations are captured. Furthermore, examination of time series of structures (e.g. figure 9) indicates that results of the k-means-based detection appear to be more continuous in time and space as well as better correlated. For example, the correlations of λ ₂ with pressure are 72 % correlated in successive time steps for clustering-based results, compared with 67 % for the conditionally sampled data. Figure 9 shows three samples of the evolution of detected QSVs colour coded by their pressure (upper row) and streamwise vorticity (lower row). As expected, the QSVs mostly appear as inclined structures that have a negative pressure, i.e. lower than the spatially averaged value in each realization (zero). Some of the structures have positive and others negative streamwise vorticity, that is, they form counterrotating vortices, consistent with the description by Bernal & Roshko (Reference Bernal and Roshko1986).

Figure 8. (a–c) Several t-SNE 2-D representations of the 45-dimensional matrix used for detecting QSVs. The colours indicate the mean-centred and variance-normalized values of the specified quantities ($\omega$_xy, λ ₂ and pressure). (d) The corresponding locations selected as QSV indicated as 1.0, and non-QSV as 0.0. (e) A comparison of correlations of negative pressure with λ ₂ and $\omega$_xy inside QSVs detected by the k means clustering and conditional sampling based on the magnitudes of λ ₂ and $\omega$_xy.

Figure 9. A sample sequence showing the evolution of detected QSVs at the indicated times colour coded by their pressure (a–c) and their streamwise vorticity (d–f).

Since the measurement resolution is limited to 200 μm, the Kolmogorov length scales are estimated based on the dissipation rate ε calculated from fitting a −5/3 slope to the inertial range of the measured temporal spectra of u ² (Liu, Meneveau & Katz Reference Liu, Meneveau and Katz1994; Pope Reference Pope2000). Figure 10 provides samples of these spectra for the two speeds averaged over points located in the middle of the shear layer and across the span. The estimated Kolmogorov scales ($\eta = {({\nu ^3}/\varepsilon )^{1/4}}$) are found to be 62 μm for 1.45 m s⁻¹ and 17 μm for 5.3 m s⁻¹, at least an order of magnitude smaller than the characteristic size of the cavities, and two orders of magnitudes smaller than the vertical extent of the shear layer. The Taylor microscales, estimated as$\sqrt {15\nu (u_i^2)/3\varepsilon } $ (Pope Reference Pope2000), are 2.1 mm at 1.45 m s⁻¹ and 0.6 mm at 5.3 m s⁻¹. Statistics on the size, strength and orientation of the QSVs are presented in figures 11 and 12. To calculate these geometric scales, one must separate structures that are connected by ‘weak’ links, defined as cases with relatively large cumulative distances from nearby non-zero voxels in all directions. Using a Matlab-based watershed transformation (Meyer Reference Meyer1994), the structures are ‘broken’ at the ridge lines, where the distributions of distance have local maxima. The length of vortices is determined based on the largest eigenvalue of the $\omega$_xy-weighted second central moments of the volume, namely the tensor ${M_{ij}} = \iint {{({x_i} - {{\bar{x}}_i})}^2} {{({x_j} - {{\bar{x}}_j})}^2}{\omega _{xy}}({x_i},{x_j})\,\textrm{d}{x_i}\,\textrm{d}{x_j} /\iint {{\omega _{xy}}({x_i},{x_j})\,\textrm{d}{x_i}\,\textrm{d}{x_j}}$. The other two eigenvalues define the cross-section and an equivalent diameter is obtained from their norm. The characteristic strength is estimated by multiplying the spatially averaged $\omega$_xy within the vortex with the cross-sectional area. The lengths and diameters of the contiguous regions of low pressures inside the QSVs are also determined by retaining regions where the pressure p falls below specified thresholds, namely C_p = p/0.5$\rho$U ² < 0, C_p < −0.1, and C_p < −0.2 and recalculating the eigenvalues of the second central moment of this volume. The 3-D orientations of the structures are based on the eigenvectors associated with the largest eigenvalues.

Figure 10. Frequency (f) spectral density of the streamwise velocity fluctuations at y/h = −0.17, averaged over the entire span and axial extent of the tomographic PIV sample volume. Dashed line is a −5/3 slope fit used for estimating η.

Figure 11. Joint probability densities in a logarithmic scale of the QSV length and diameter at 1.45 m s⁻¹ (a–c), and 5.3 m s⁻¹ (d–f) for: (a,d) QSVs, (b,e) negative pressure (C_p < 0) regions within the QSVs and (c,f) low pressure (C_p < −0.1) regions within the QSVs. Black and red lines indicate aspect ratios of 2 and 5, respectively. (g,h) Conditional PDFs, at the velocities indicated in the legend, of the: (g) diameters, and (h) aspect ratios of the QSVs, low pressure regions in them and the cavities.

Figure 12: Comparisons between PDFs of the separation between two adjacent low pressure regions, conditioned on C_p < −0.1, at U = 1.45 and 5.3 m s⁻¹, and the distance between two cavitation events along the same QSV at U = 10.5 and 16 m s⁻¹.

The most likely diameter of the $\omega$_xy weighted QSV is 1.25 mm and its length is 2 mm for both speeds (figure 11a,d). However, a significant fraction of the data has aspect ratio falling between 2 and 5, with diameters extending to 4 mm, albeit with low probability. With increasing speed, the fraction of structures with diameters exceeding 2 mm and lengths over 10 mm increases. As expected, the regions with C_p < 0 (figure 11b,e) and C_p < −0.1 inside the QSVs (figure 11c,f) are smaller than the entire QSVs. For both thresholds, the most probable low pressure zones have a diameter of approximately 0.2 mm, and an aspect ratio of around 2, but values extend to 5 at both speeds. However, at the higher speed there are more QSVs with diameters exceeding 2 mm and aspect ratios larger than 2 (figure 11a,d) as well as structures of the same size with negative pressures (figure 11c,f). These trends are consistent with those shown by the cavities in figure 7(a). Also like the cavities, the largest aspect ratio (~5) occurs for structures with diameters up to about half of the maximum. However, the different trends with velocity vanish in the C_p < −0.1 plots (figure 11c,f). The PDFs comparing the diameters and aspect ratios of cavities with those of the QSVs and the low pressure regions within them are presented in figures 11(g) and 11(h), respectively. The cavity diameters are smaller than those of the QSVs, as one would expect, consistent with trends observed previously for tip vortices (Choi & Ceccio Reference Choi and Ceccio2007; Pennings, Westerweel & van Terwisga Reference Pennings, Westerweel and van Terwisga2015). As the pressure thresholds are lowered, the most probable diameters of low pressure zones shift from 1.2 mm, the characteristic size of the entire QSVs, to 0.2 mm, matching the characteristic diameters of the cavities at both velocities. The low-probability largest diameters also approach those of the cavities with decreasing pressure threshold, matching them for C_p = −0.2. The most probable aspect ratios of the cavities are smaller than those of the QSVs (figure 11g) owing to the presence of the ‘small spherical’ cavities during the initial growth phase. On the high end, the aspect ratios of the cavities are larger than those of the QSVs. However, as the pressure threshold is lowered, the aspect ratios of the low pressure regions extend to higher values, nearly matching those of the cavities at 16 m s⁻¹, and exceeding those at 10.5 m s⁻¹. Furthermore, statistics of the distance between pressure minima along the QSV, conditioned on C_p < −0.1, are compared with the distance between cavitation events along the vortex in figure 12. The most likely separation between events is around 0.2h of either the pressure minima or the cavities, consistent with the sample visualizations in figures 6 and 9, although the pressure-based modes are slightly smaller. The peaks are broad, and a significant fraction of the data extend to 0.6h, where both PDFs increase with velocity. In summary, the dimensions of the cavities and the distance between them match those of the low pressure regions within the QSVs. These agreements provide strong evidence that the spatial distribution of the early cavitation events is influenced significantly by the pressure field.

Figure 13(a) presents the normalized strength of the vortices. The most probable structures have a strength of 0.06Uh at both speeds. While most of the PDFs for the two speeds collapse, they diverge at strengths exceeding 0.4Uh, where the probability increases with velocity. The characteristic strength of spanwise vortices can be estimated as Uδ_w, where δ_w is the vorticity thickness (Brown & Roshko Reference Brown and Roshko1974), defined as $U/{(\partial u/\partial y)_{max}}$. Based on the mean velocity distributions, at the present site δ_w = 0.65h and 0.7h for U = 1.45 and 5.3 m s⁻¹, respectively. Hence, the most probable strength of the QSVs is approximately 10 % of those of the spanwise vortices, consistent with previously reported levels (Bell & Mehta Reference Bell and Mehta1992). However, the extreme values, representing e.g. less than 0.1 % of the structures, have strengths (0.65Uh at 1.45 and 0.74Uh at 5.3 m s⁻¹) that are comparable to those of the spanwise structures. As noted above, this is the range where the probabilities are velocity dependent (figure 13a). Statistics on the orientation of the QSVs with aspect ratios over 2 are presented in figure 13(b) as the angle with the laboratory coordinate system. Results for the two velocities appear to be remarkably similar. The angles with respect to the x directions are preferentially small and most likely to be 15°–30°. The structures are likely to be perpendicular to the spanwise directions, and to a lesser extent to wall-normal directions as well. Applying pressure thresholds does not change the trends and are, therefore, not presented. The corresponding statistics of the orientation of the cavities in the region of tomographic PIV data are presented in figure 13(c). Their preferred orientations include small angles to the streamwise directions with a peak around 30°. Like the QSVs, close to perpendicular directions to the spanwise are preferred. However, the cavities have with almost zero probability of being vertical or <30° with preference for angles of 60–70°, remaining significant up to 90°.

Figure 13. The PDFs of the QSV: (a) strength, and (b) angles (in degrees) with the x, y and z directions at U = 1.45 and 5.3 m s⁻¹; and (c) PDFs of the corresponding angles of the cavities.

The statistics of pressure in the shear layer are provided in figure 14. Figure 14(a) compares the PDFs of pressure inside and outside the QSVs, with the latter representing about 83 % of the entire data. All the PDFs are nearly symmetric, with the tails being more intense than Gaussian distributions with same mean and standard deviation (also shown). These trends are consistent with previously published data on pressure fluctuations in turbulent shear flows (Brandao & Mahesh Reference Brandao and Mahesh2022). In contrast, for isotropic turbulence, the PDFs are non-symmetric with longer negative tails, which increase with Reynolds number (Bappy et al. Reference Bappy, Carrica and Buscaglia2019). The pressures are lower inside the QSVs, with similar shifts for both velocities, generally scaling with U ². Consequently, a pressure coefficient of −0.2 is over 7 times more likely to appear inside the QSV than outside of this vortex. The tails of the pressure fluctuations both inside and outside of the QSVs are larger at 1.45 m s⁻¹ than those at 5.3 m s⁻¹. Based on the trends of inception indices in shear layers reported in Katz & O'Hern (Reference Katz and O'Hern1986), the inception indices at the current Reynolds numbers of velocity measurements are expected to fall in the 0.2 to 0.4 range, in agreement with the magnitudes of the present pressure minima. Recall that the present cavitation inception tests with indices in the 0.5 to 0.6 range are performed at higher Reynolds numbers. These values are also in agreement with the trends reported in Katz & O'Hern (Reference Katz and O'Hern1986). Moreover, the magnitudes of pressure minima are consistent with those of Rankine vortices with the same size and strength as those of the QSVs. For example, a Rankine vortex consisting of a solid body rotation core with diameter D of 1.2 mm and strength $\varGamma$ = 0.06Uh surrounded by an irrotational flow, has a core pressure coefficient (pressure $\textrm{drop} = \rho {(\varGamma /{\rm \pi} D)^2}$) of −0.17. Furthermore, using C_p = −0.3 to represents the pressure minimum in the core of QSVs at 5.3 m s⁻¹, following Choi & Ceccio (Reference Choi and Ceccio2007), the expected growth rate of cavities along the QSV axis should be 1.65U. This value agrees with the presently measured rates at 10.5 and 16 m s⁻¹ (figure 7c). Figure 14(b) provides the temporal pressure spectra at y/h = −0.17, averaged over the streamwise and spanwise directions. At both speeds, the spectra peak at Strouhal numbers of 0.2–0.3, as discussed in several previous studies, including Driver et al. (Reference Driver, Seegmiller and Marvin1987), which relate it to flapping of the shear layer. In the inertial region beyond this peak, the spectral decay has a slope of −1.6, consistent with that of a turbulent boundary layer (Tsuji et al. Reference Tsuji, Fransson, Alfredsson and Johansson2007). Additionally, at 5.3 m s⁻¹, there are a few low-frequency peaks, which Rockwell & Naudascher (Reference Rockwell and Naudascher1979) have attributed to intermittent switching of oscillatory modes based on kinetic energy spectra.

Figure 14. (a) The PDFs of the pressure inside and outside the QSVs. Black curves are Gaussian distributions with the same mean and standard deviation as those of the pressure at U = 5.3 m s⁻¹. (b) Pressure spectra averaged over the spanwise and streamwise directions at y/h = −0.17.

To evaluate the effect of vortex stretching on the pressure, the statistics of the stretching term along the vortex axis are presented in figure 15. Vortex stretching has been associated with the energy cascade and intermittent formation of regions with high velocity gradients (Taylor Reference Taylor1938; Buaria, Bodenschatz & Pumir Reference Buaria, Bodenschatz and Pumir2020) and formation of pressure minima in vortices (Chang et al. Reference Chang, Choi, Yakushiji and Ceccio2012). Figure 15(a) presents PDFs of $\varPhi$ inside and outside of the QSVs. Stretching (and contraction) inside the QSVs are much higher compared with those in the surrounding at both speeds, and once scaled with ${(U/h)^2}$, results collapse. Stretching events ($\varPhi$ > 0) are more frequent than contractions everywhere, a typical feature of turbulent flows owing to buckling of vortices during contraction (Rogers & Moin Reference Rogers and Moin1987). Nonetheless, the QSV contains regions of both significant compression and stretching, consistent with Verzicco & Jimenez (Reference Verzicco and Jimenez1999). Figure 15(b) presents PDFs of the correlations between pressure and the stretching term at the same time and location, with the pressure conditioned on being negative and the stretching on being positive (stretching). Clearly, the PDFs corresponding to the interior of the QSVs are substantially broader than those of the surrounding flow, indicating that in some cases, pressure minima co-occur with high stretching, as one would expect, yet in other cases, they are not. These findings have led to a detailed analysis of the evolution of pressure and stretching inside the QSVs and the relations between them based on Lagrangian statistics. Figure 15(c) shows PDFs of the axial strain rate magnitude inside the QSVs. Similar to other ensemble-averaged turbulence statistics, the vortex axial straining magnitudes scale with U/h, with the most probable values fall in the U/h to 2U/h range. These magnitudes are consistent with the shear strain along the periphery of spanwise vortices with strength of 0.65Uh, and radius of 0.32h, hence core azimuthal velocity of 0.32U and strain rate of U/h.

Figure 15. (a,b) The PDFs of the: (a) vortex stretching term, and (b) correlations between negative pressure and positive stretching; (c) PDFs of the axial strain rate in the QSVs.

Statistics on the instantaneous alignment of the vorticity relative to the principal strain rate directions are presented in figure 16(a,b). As shown in previous studies (Ashurst et al. Reference Ashurst, Kerstein, Kerr and Gibson1987; Tsinober, Kit & Dracos Reference Tsinober, Kit and Dracos1992), the vorticity is preferentially aligned with the intermediate strain rate. This phenomenon is slightly more pronounced inside the QSVs, consistent with the previously observed increase in alignment with the intermediate strain, as the vorticity magnitude increases (Huang Reference Huang1996; Buaria et al. Reference Buaria, Bodenschatz and Pumir2020). There is no significant alignment preference with the most extensive strain, and the vorticity is preferentially perpendicular to the most compressive eigendirection with a slight increase inside the QSVs. Next, we examine the alignment of the QSV structure relative to the principal strain rate directions (figure 16c) and the vorticity (figure 16d). Since the QSV structure is often curved, its alignment is defined based on the direction of the largest eigenvector of M_ij introduced previously. Figure 16(c) shows that the QSVs are more likely to be aligned with the most extensive strain rates eigenvector and less likely to be aligned with the intermediate strain direction in comparison with the trends of the vorticity (figure 16a,b). Yet, overall, the QSVs are still preferentially aligned with the vorticity (figure 16d), with the distribution being quite broad. In terms of structure, the PDFs of the strain state parameter (figure 17) confirm the previously published preference for axisymmetric extension (Lund & Rogers Reference Lund and Rogers1994), i.e. a pancake shape. However, this preference is lower inside the QSVs, implying that they contain more axisymmetric contraction events, as one would expect for stretched vortices. All the alignment and configuration trends at the two velocities are remarkably similar. The evolution of s* with time in the same vortex is discussed later based on Lagrangian statistics.

Figure 16. The PDFs of the cosines of the angles between the: (a) vorticity and the principal strain rates outside, and (b) inside the QSVs; (c) the QSV axes and the local principal strain rates, and (d) the QSV axes and the local vorticity. Here, s ₁, s ₂ and s ₃ are the most extensive, intermediate and most compressive strain rate eigenvectors, respectively.

Figure 17. The PDFs of the strain state parameter (Lund & Rogers Reference Lund and Rogers1994). As illustrated, 1 indicates a ‘pancake’ structure, and −1, a stretched vortex.

Next, we examine the impact of viscous diffusion in a direction perpendicular to the vortex axis on the evolution of the QSVs. The ratio of viscous diffusion $(\nu {\nabla ^2}\omega )$ to axial vortex stretching terms, which are presented in figure 18(a), is typically smaller than one. For the majority of the cases, as long as one does not consider the temporal variations in the strain field (discussion follows), the impact of viscous diffusion is significantly smaller than that of the vortex stretching, especially at 5.3 m s⁻¹. However, in a few cases, the two terms become comparable, much more so at 1.45 m s⁻¹. Furthermore, the present structures are very rarely stretched to the viscous diffusion limit, referring to a situation where the viscous diffusion is equal to the stretching, hence the vortex core cannot shrink owing to stretching. For an axisymmetric vortex, such situation occurs when $4\nu U/hD_\mu ^2 = \varPhi$, where D_μ is the minimum diameter. Figure 18(b) provides the PDF of the instantaneous ratio of D_μ to the actual vortex diameter for the entire QSV data. As is evident, the stretching limit is rarely reached. Note, however, that the relative significance of viscous effects increases with decreasing Reynolds number.

Figure 18. The PDFs of the ratio of the: (a) radial viscous vorticity diffusion in the QSVs to the axial stretching, and (b) the diameter at the axial stretching limit to the measured QSV diameter.

Lagrangian statistics: the analysis in a Lagrangian framework examines the durations and sequence of events underlying the development of pressure minima. With this aim, 10⁵ synthetic, neutrally buoyant, initially randomly distributed particles are tracked in time for the entire 2 s duration of data acquisition. Their trajectories are determined by using a fourth-order Runge–Kutta integration of the local velocity. Particles that leave the field of view are re-seeded at the upstream boundary of the sample volume, while ensuring there is no bias in their spatial distributions by diluting regions with increased concentration of particles. To investigate how long the low pressure events last, figure 19 shows the fraction of time all the particles experience pressures below the specified threshold levels plotted versus their dimensionless durations. This procedure is similar to the analysis reported in Bappy et al. (Reference Bappy, Carrica and Buscaglia2019) based on DNS data of isotropic turbulence. The experimental durations are truncated by the residence time of particles being translated through the sample area at the free-stream velocity, namely 1.2h/U. In figure 19(a) the results inside and outside of the QSVs for the higher velocity are compared, and figure 19(b) shows the effect of velocity for the interior of QSVs. The dotted vertical lines mark the measured initial cavitation growth time, ~0.2h/U, samples of which are presented in figure 7(b). As expected, the time fractions decrease with decreasing pressure threshold. For the same speed, the low pressure events inside the QSVs last significantly longer than those outside of them (figure 19a). Some of the durations are longer than the initial growth time, but most of them (~80 %) are not. Trends in the effect of velocity appear to depend on the threshold level. For a mild pressure threshold (−0.05 and −0.1), the time fractions are higher at 5.3 m s⁻¹ than those at 1.45 m s⁻¹. However, as the threshold is reduced to –0.15 the difference between them diminishes. For all thresholds, the slopes are persistently milder at the higher velocity, implying that the long duration events are more likely to occur at 5.3 m s⁻¹. Bappy et al. (Reference Bappy, Carrica and Buscaglia2019) also show that trends of the frequency of low pressure events with Reynolds number reverse with increasing duration for isotropic turbulence. They decrease with Re for short durations, consistent with the present findings, but they increase for long durations. Unfortunately, we cannot increase the present range of durations without increasing the field of view, hence compromising spatial resolution.

Figure 19. The time fraction of the indicated low pressure events and their durations: (a) inside and outside of the QSVs at 5.3 m s⁻¹, and (b) inside the QSVs for the two speeds. The dotted vertical lines indicate the scaled initial growth times of the cavities.

The Lagrangian correlation of pressure at time t with various quantities at time t + Δt for both speeds, inside and outside of QSVs, are provided in figures 20 and 21. Figure 20(a) shows that the unconditioned pressure–pressure correlations remain above 0.4 for 1.45 m s⁻¹ and 0.5 for 5.3 m s⁻¹, for durations up to 0.8h/U. However, the slopes of all the correlations indicate a more rapid decay for durations of less than 0.2h/U. At longer times, the correlations inside and outside the QSVs start to deviate, with those outside of the QSVs decreasing at a faster rate compared with those inside. Figure 20(b) provides the pressure–pressure correlations conditioned on C_p < −0.05 at both times. These correlations decay at faster rates than the unconditioned results, and the differences between the interior and exterior of the QSVs start at Δt = 0. These trends are consistent with the longer durations of low pressure events inside the QSVs (figure 19a). Here again, the decay rates of the correlations become distinctly slower at Δt > 0.2h/U. For the most part, the correlations increase with velocity, especially at short times. At Δt<0.2h/U, the differences are significant, but they diminish at Δt > 0.8h/U. The differences inside the QSVs are smaller in the conditional statistics (figure 20b), also consistent with the trends depicted in figure 19(b). In summary, the pressure–pressure correlations decrease at a faster rate for short durations, during which the correlation increase with velocity. The variations with velocity diminish for longer times.

Figure 20. Lagrangian pressure–pressure correlations inside and outside the QSVs at the indicated speeds for: (a) the entire data, and (b) conditioned on C_p < −0.05.

Figure 21. Lagrangian correlations of pressure at time t (conditioned on C_p < −0.05 and $\omega$_xy > 5U/h) with the following quantities at time t + Δt: (a) $\omega$_xy (b) axial vorticity stretching and (c) contraction inside and outside of QSVs. (d) The average strain state parameter at time t + Δt.

Figure 21(a) provides the pressure–$\omega$_xy correlations conditioned on C_p < −0.05 and $\omega$_xy > 5U/h for positive and negative Δt. The correlation magnitudes are generally low (<0.2), but are higher inside the QSVs. As expected, they peak at Δt = 0, indicating that low pressure events preferentially co-occur with elevated $\omega$_xy only inside the QSVs. While the peak value at 5.3 m s⁻¹ is lower than that at 1.45 m s⁻¹, the correlations at 5.3 m s⁻¹ remain elevated for a longer duration, implying that the pressure minimum and (presumably associated) vorticity maximum last for a longer time. The correlations of low pressures with axial vorticity stretching and contraction events are shown in figures 21(b) and 21(c), respectively. Outside the QSVs, the correlations remain low irrespective of time. However, inside the QSVs the correlations with contraction peak at Δt  = 0.2h/U and those with stretching between −0.2h/U and −0.8h/U. These trends suggest that low pressure events in the QSVs preferentially involve a dynamic process consisting of a period of stretching with duration of about 0.6h/U, followed by a period of low pressure, ending with a contraction event. Conditional sampling based on the evolution of straining (not shown) confirms that the pressure decreases during the period of peak straining, reaching minimum when as the straining diminishes.

The Lagrangian evolution of average (over all particles) strain state parameter when the pressure coefficient at time t is lower than −0.05 are presented in figure 21(d). As shown earlier, the QSVs have lower values of s* compared with the surrounding flow, corresponding to a reduced tendency for axisymmetric extension. Starting from Δt ~ −0.6h/U, around the time where the vortex stretching starts (figure 21b), s* decreases, reaching a minimum at Δt ~ 0.2h/U, at the time of peak contraction. These observations suggest causal relationships between vortex stretching and increased tendency towards axisymmetric contraction inside the QSVs. The same trends, but with persistent higher s*, are observed in the surrounding flow. Before concluding this section, one should note that in contrast to the pressure–pressure correlation, all the Lagrangian correlations presented in figure 21(a–c) are more prominent at 1.45 m s⁻¹ compared with those at 5.3 m s⁻¹. The primary cause involves local fluctuations in the magnitudes of vorticity and stretching, which are larger at the high velocity, presumably owing to the higher turbulence level. When the vorticity and stretching term are spatially filtered using e.g. a 3-D Gaussian filter with standard deviation of 0.2 mm, the correlation magnitudes increase, and the differences in correlation between the two speeds diminish, as illustrated in figure 22. The time delay of the peaks in the stretching–pressure and contraction–pressure correlations are longer at 5.3 m s⁻¹, consistent with the longer durations of the vorticity–pressure correlations (figure 22b).

Figure 22. Lagrangian correlations of pressure at time t (conditioned on C_p < −0.05 and $\omega$_xy > 5U/h) within the QSVs with the following spatially filtered quantities at time t + Δt: (a) axial vorticity stretching, (b) $\omega$_xy and (c) axial contraction. The spatial filter involves a 3-D Gaussian filter with standard deviation of 0.2 mm.

The time scales of minimum pressure and associated stretching and peak vorticity events are examined next. Figure 23 provides the changes to the magnitude of the average pressure over time, ΔC_p = C_p(t + Δt) − C_p(t), as well as the corresponding average stretching term $\varPhi$ (positive or negative) and vorticity $\omega$_xy for the same particle group within the QSV. Errors estimated based on bootstrapping by dividing the data into 5 sets are also presented. Similar to the correlations, the data are provided for when a pressure minimum for each particle trajectory occurs at Δt = 0. Only cases where minimum pressure coefficient is below −0.05 and instantaneous $\omega$_xy exceeds 5U/h are considered. The average pressure minimum at 1.45 m s⁻¹ is deeper compared with that at 5.3 m s⁻¹, but this minimum develops and decays at a significantly faster rate. The time scale of the exponential decay rate is 0.11h/U at 1.45 m s⁻¹ vs. 0.28h/U at 5.3 m s⁻¹. These trends with velocity, and the magnitudes of time scales, are consistent with those of the initial decrease in pressure–pressure Lagrangian correlations (figure 20b), i.e. 0.2h/U. While the pressure-conditioned average stretching is always positive, it peaks at Δt ~ −0.5h/U, and then decreases monotonically until Δt ~ +0.3h/U, the timing of peak conditional contraction. The duration of transition is clearly longer than that of the change in pressure. Another result of note is the time delay between two stretching peaks, ~1.5h/U (figure 23b), which appears also in the Lagrangian pressure–stretching and pressure--contraction correlations (figure 22a,c, respectively). The average vorticity peaks when the pressure is minimum and the trends of vorticity are nearly identical for the two speeds. In summary, it appears that the dynamic changes to the vortex pressure and strain fields involve two time scales, a shorter time over which the pressure decays, and a longer time consistent with the stretching–contraction cycle. While the latter is independent of the Reynolds number (figure 23b), the shorter time scales of pressure decay increase with velocity.

Figure 23. Lagrangian evolution of: (a) pressure change, (b) stretching and (c) vorticity when C_p is minimal at Δt = 0 (conditioned on C_p < −0.05 and $\omega$_xy > 5U/h) at U = 1.45 and 5.3 m s⁻¹. The error is determined from a bootstrap analysis involving 5 different subsets.

The next discussion describes our attempt to understand the flow mechanisms affecting the spatio-temporal evolution of pressure and straining in the QSVs. The following analysis, therefore, describes the length scale of events along the QSV axis. To find the QSV axis, after applying the watershed transformation as described before, the vortex is skeletonized (Lee, Kashyap & Chu Reference Lee, Kashyap and Chu1994), i.e. reduced to a line or line segments defining its centre, which is not necessarily the centre of vorticity. This step is only used for identifying the ends of the vortex segment, and for determining whether the vortex contains multiple branches. Then, starting from the detected ends in the case of multiple branches or the centroid in the case of a single branch, the ‘vortex axis’ is built by marching by a distance of 100 μm along the local (total) vorticity vector until a boundary of the structure is reached. This process is repeated for each vortex segment. This axis is evolved in time by displacing synthetic particles located less than 50 μm away from the axis according to the local velocity to determine its next location. The result is compared with the location of the detected QSV axis in the next frame. Due to limitations in spatial resolution, the displaced axis based on particle motion and that determined based on the vorticity distribution could be different by up to the grid size of 200 μm. Only particles located less than 100 μm from the detected axis in the next frame (overwhelming majority of them) are used for defining the location of the QSV in the next time step. In cases of discontinuities, the vortex axis in the next frame is used for filling missing segments. Vortex axis locations that do not correspond to segments detected in previous times are treated as new structures, which are then allowed to evolve in time. Figure 24 shows several sample QSV along with their detected centres. The pressure, strain rate, axial velocity, vorticity and vorticity stretching terms along the axes are recorded. Approximately one half of the approximately 60 000 detected QSVs for each velocity are rotating along the flow direction, i.e. ω_x is positive, and the other half are opposite to it.

Figure 24. Samples of QSV segments as determined by the k-means clustering (in red) along with their detected axes.

Figure 25 shows the conditionally averaged evolution of pressure and vortex stretching along the same QSV axis at five time steps around a pressure minimum detected at dt = 0. For each time (two with dt < 0 and two with dt > 0, and for dt = 0), we provide the distribution of average pressure and stretching along the axis of the same (displaced) vortex. That means that for all the curves dl defines the distance from the same segment along the vortex axis where the pressure minimum is located at dt = 0. Hence, the displacement of the entire curve represents e.g. propagation of a pressure wave along the vortex axis. The data are conditioned on pressure minima with C_p < −0.05 as well as $\omega$_xy > 5U/h at dt = 0 and ds = 0. Results for the two velocities are presented separately and, owing to the differences in dimensionless time resolutions of data acquisition, the time steps, i.e. values of dtU/h, are different. Both clearly show evidence of a standing wave of vortex stretching. The evolutions of the stretching terms (figure 25b,d) indicate higher than the conditionally average values at dt < 0 (the average is 2.2U ²/h ² at 1.45 m s⁻¹ and 2U ²/h ² at 5.3 m s⁻¹), and lower values at dt > 0. These trends are consistent with the results of Lagrangian correlations in figures 21 and 22, which show that the pressure minimum is preceded by a stretching event and followed by a contraction. Based on figure 25, the characteristic wavelength λ of the stretched vortex segment is approximately 2.4 mm at both speeds, approximately twice the most probable diameters of the QSVs (figure 11). Recall that the typical aspect ratio of the QSVs is also approximately 2 (figure 11) In comparison, the Taylor microscales at 1.45 and 5.3 m s⁻¹ are 2.1 and 0.6 mm, respectively. Therefore, the size of the stretched vortex sections is closer to the dissipation scales at 1.45 m s⁻¹. This observation introduces the possibility that differences in viscous effects influence the time scales of low pressure events, as elaborated upon in the discussion that follows. The time variations of pressure decay are the same as observed in figure 23(a).

Figure 25. (a,c) The pressure and (b,d) axial vortex stretching along the QSV axis at the indicated successive times, when the pressure is minimal at dt = 0 and dl = 0 (conditioned on C_p < −0.05 and $\omega$_xy > 5U/h). (a,b) U = 1.45 m s⁻¹ and (c,d) U = 5.3 m s⁻¹.

Elucidating scale effects in QSV dynamics based on the data: in an attempt to elucidate the mechanisms affecting the observed time evolution of low pressure events within the QSVs, we rely on previously published data on the dynamics of deformed vortices (Verzicco & Jimenez Reference Verzicco and Jimenez1999; Pradeep & Hussain Reference Pradeep and Hussain2001; Abid et al. 2002; Moet et al. 2005). Conceptual schematic descriptions of the processes occurring when a segment of a vortex undergoes local stretching are illustrated in figure 26. When the core size and pressure decrease locally (figure 26b), a part of the initially axial vorticity realigns in the azimuthal direction (Abid et al. 2002). In a cylindrical coordinate system depicted in figure 26, this realignment corresponds to the $\partial {\omega _\theta }/\partial t = {\omega _z}\partial {v_\theta }/\partial z$ term in the vorticity transport equation. This process in effect generates two vortical rings on both sides of the stretched area, or what has been referred to as coiling of vortex lines. Flow induced by the azimuthal rings opposes the initial stretching, eventually causing a contraction (figure 26c). In extreme cases, collision between these opposite sign vortex rings can cause vortex bursting (Verzicco & Jimenez Reference Verzicco and Jimenez1999). Once contracted, a new set of rings forms, which reverses the axial contraction, resulting in an oscillatory motion of stretching and contraction, hence periodic formation of low pressure regions. According to simulations by Abid et al. (2002), the older rings do not decay, but new successive ones with alternating signs form inward of the previous rings. Therefore, the new rings have a stronger effect on the stretching or compression of the axial vortex than the previous generations. Multiple sites of local stretching–contraction cycles could presumably create intermittent islands of low pressure along the vortex, a scenario consistent with the multi-point cavitation events shown in figure 6. The vortex also diffuses, and consequently both the amplitude of its oscillations and its overall strength decay under the influence of viscosity, with the rate of diffusion decreasing with increasing Reynolds number (Verzicco & Jimenez Reference Verzicco and Jimenez1999; Abid et al., 2002). Abid et al. (2002) also show that the vortex response to axial straining is primarily an inertial process, with the frequency of oscillations f, being dependent only on the initial vortex radius R ₀, vorticity $\omega$₀ and the wavelength of the stretching events λ, namely f = 2.4$\omega$₀R ₀/λ. For the present conditions, using a vortex radius of 0.06h (figure 11g) to represent the vortex prior to local stretching, a strength of 0.06Uh (figure 13a), and length of stretching event of λ = 0.24h (figure 25), the predicted frequency is 3.7U/h, i.e. a period of 1.7h/U. This value is consistent with the presently measured period of stretching–contraction events presented in figure 23(b), suggesting that the observed processes are indeed a result of oscillatory vortex reaction to local stretching events.

Figure 26. A schematic showing the evolution of a vortex segment subjected to local stretching: (a) prior to stretching, (b) formation of secondary azimuthal rings that oppose the stretching and cause contraction and (c) formation of rings that oppose the contraction and cause extension.

The next discussion attempts to estimate the changes to the vortex radius as the pressure evolves around the time of minimum pressure. Unfortunately, the present spatial resolution, 0.2 mm, is not sufficient for resolving the time evolution of the vortex radius R during the stretching-contraction events. However, the evolution can be deduced from the pressure measurements. For example, in a Rankine vortex consisting of a rotational core with radius R and a uniform vorticity, the pressure in the centre is $p = {p_\infty } - 5\rho {\varGamma ^2}/8{{\rm \pi} ^2}{R^2}$. Hence,

(3.2)\begin{equation}\varDelta {C_p}/{C_{p0}} + {C_{p\,min}}/{C_{p0}} = {({R_0}/R)^2},\end{equation}

where C_{p 0} is the conditionally averaged pressure coefficient before the stretching starts, R ₀ is the corresponding initial radius, assumed to be 0.6 mm for the two speeds, ${C_{p\,min}}$ is the minimum conditionally averaged pressure coefficient and values of ΔC_p are provided in figure 23(a). The present values are: C_{p 0} = −0.04 and −0.02, and ${C_{p\,min}} ={-} 0.16$ and −0.11 at 1.45 and 5.3 m s⁻¹, respectively. Equation (3.2) allows us to estimate the time evolution of R/R ₀ using the measured pressure. The results for both velocities, assuming that stretching begins at Δt = −1.3h/U, are presented in figure 27. It shows that the core radius is almost halved during stretching, decreasing to a size consistent with the core diameter during low pressure events (0.4 mm, see figure 11g). The evolution of the radius can also be evaluated for a vortex subjected to straining and viscous diffusion. The viscosity-induced time evolution of circulation in a Lamb–Oseen vortex is $\varGamma = {\varGamma _0}(1 - {\textrm{e}^{ - {r^2}/4\nu t}})$, where $\varGamma$₀ is the vortex strength far from the core (Batchelor Reference Batchelor1967). For this vortex, the radius of maximum tangential velocity is $R(t) = 2.24\sqrt {\nu t} $, i.e. ${R^2}(\varDelta t + 1.3h/U) = R_0^2 + 5.02\nu (\varDelta t + 1.3h/U)$. The viscous terms are significant, e.g. the viscous growth during a period of h/U is 31 % and 16 % of R ₀ for 1.45 and 5.3 m s⁻¹, respectively. In addition to the viscous effects, the radius also changes due to vortex stretching and contraction. For a cylindrical vortex element with a constant volume of ${\rm \pi} {R^2}\lambda $, $\textrm{d}R/\textrm{d}t ={-} sR/2$, where $s = 1/\lambda \,\textrm{d}\lambda /\textrm{d}t$ is the axial strain rate. Thus, the axial strain causes an exponential rise or decay of the vortex core radius. Accounting for both straining and viscous diffusion, an approximation for the growth rate of the radius is

(3.3)\begin{equation}\textrm{d}R/\textrm{d}t ={-} sR/2 + 2.5\nu /R,\end{equation}

which, for a fixed s over time would yield,

(3.4)\begin{equation}{(R/{R_0})^2} = (1 - 5.02\nu /sR_0^2)\,{\textrm{e}^{ - s(\varDelta t + 1.3h/U)}} + 5.02\nu /sR_0^2.\end{equation}

The time evolutions of R/R ₀ for characteristic stretching with a strain rate of s = U/h (figure 15c) for a duration of 1.3h/U at both speeds, followed by contraction with the same magnitude and duration, are also plotted in figure 27. As time increases, the changes to the radius at the two speeds diverge owing to differences in viscous effects. The viscosity slows down the reduction in radius during stretching, and enhances its growth rate during contraction. The magnitude and trends with velocity of the pressure-based estimates, and those calculated for the strained viscous vortex, are similar. Differences can be attributed e.g. to our use of a fixed strain rate magnitude throughout the vortex evolution, as opposed to the time-varying s, uncertainty in R ₀ and pressure and much higher complexity in the actual system. Results obtained when the viscous term in (3.4) is removed, i.e. for ${(R/{R_0})^2} = {\textrm{e}^{ - s\varDelta t}}$, also shown in figure 27, are identical for the two speeds. As expected, the inviscid vortex has a smaller radius than the viscous ones. Note that the modelled trends of the QSV radius with Reynolds number are consistent with those of the measured values (figure 11g). While the unconditioned diameters at the two speeds are similar, those of low pressure events decrease with increasing velocity. In summary, this analysis suggests that, despite the nearly identical (scaled) straining and initial QSV diameter, viscous diffusion causes significant differences in the evolution of vortex radius at the two speeds. To understand this trend, note that while the overall Reynolds number of the current shear layer is high, at the scales of the QSVs, the Reynolds numbers are significantly lower, e.g. $\varGamma$/ν is 870 and 3180 at 1.45 and 5.3 m s⁻¹, respectively.

Figure 27. A comparison of the Lagrangian evolution of the QSV radius based on the measured evolution of pressure when C_p is minimal at Δt = 0 ((3.2), dashed lines) with that of a viscous vortex subjected to stretching followed by contraction, both with a magnitude of |s| = U/h (3.3).

The minimum pressure in the modelled stretched vortex decreases with increasing Reynolds number. For example, for the same initial pressure coefficient of −0.03 (mode of the PDF in figure 14a), the modelled minimum pressure coefficient for s = U/h for a duration of 1.3h/U is −0.09 and −0.1 at 1.45 and 5.3 m s⁻¹, respectively. Since the turbulence, strength of vortices and strain rate scale with U (figures 4, 13a and 15c), it is reasonable to assume that the same trends hold for higher speeds. Accordingly, the characteristic minimum pressure coefficient in the stretched vortices is −0.121 and −0.122 at 10.5 and 16 m s⁻¹, respectively, i.e. the Reynolds number effects diminish under these conditions. To reach the state of early cavitation events at 10.5 and 16 m s⁻¹, e.g. σ = 0.5 (figure 5b), which would require ${C_{p\,min}} \le - 0.5$, a vortex with the same size and initial pressure has to be subjected to a stretching strain rate of s = 2.2U/h. Based on figure 15(c), such straining is ubiquitous, hence stretching to the cavitation inception levels is consistent with the present data. The modelled time scales associated with low pressure events can also be estimated by relating the radius to C_p (3.2), and using (3.4) for calculating the time required for the (negative) core pressure coefficient to increase from its minimum value to half that minimum during contraction. For the conditions presented in figure 27, this time is 0.34h/U and 0.52h/U at 1.45 and 5.3 m s⁻¹, respectively. These values are of the same order of magnitude but larger than the corresponding experimental ones, 0.12h/U and 0.3h/U, based on the data in figure 23(a). Furthermore, the predicted and measured time scales are consistent with the conditional statistics of durations presented in figure 19. For a threshold C_p of −0.05, similar to the data in figure 23(a), for the same fraction of time (e.g. 2 × 10⁻⁴), the duration at 5.3 m s⁻¹, 0.52h/U, is longer than that at 1.45 m s⁻¹, 0.34h/U (figure 19b). The longer durations with increasing Re are also a likely contributor to the higher Lagrangian pressure–pressure correlations at 5.3 m s⁻¹ (figure 20). Finally, extrapolating the modelled trends to the cavitation speeds, the time scales of low pressure events at 10.5 and 16 m s⁻¹ are 0.56h/U and 0.58h/U, respectively, i.e. they are much closer than those of the pressure measurements. If we extrapolate the experimental durations instead, but still maintain the logarithmic trend predicted by the model, the gap would be significantly bigger, 0.48h/U and 0.62h/U, respectively. In both cases, these time scales are longer than those of the duration of cavitation events, ~0.2h/U (figure 19). With increasing durations of low pressure events, one would expect an increase in the likelihood of cavitation events (figure 5b), and once they occur, longer cavities, consistent with figure 11(h).

3.4 Nuclei dynamics

This section examines how the nuclei spatial distributions with respect to the low pressure regions affect cavitation inception. Figure 28(b–d) provides the number density of bubbles for ‘natural’ seeding for the locations indicated in figure 28(a) at various cavitation indices. The free-stream distributions are quite similar at all speeds and σ, except one case, for U = 16 m s⁻¹ and σ = 0.45, which has a similar distribution, but slightly higher concentrations. These size distributions are also similar in magnitude and slope (−4) to those compiled from prior laboratory and oceanic observations (e.g. Khoo et al. Reference Khoo, Venning, Pearce, Takahashi, Mori and Brandner2020). There are many possible reasons for the slight increase in the highest speed and lowest cavitation index, such as some minor cavitation elsewhere in the test facility. In the recirculation zone, for each velocity, the number density increases with decreasing σ, and the slopes are steeper, indicating higher fractions of small bubbles compared with large ones. Most notable, the concentrations at 10.5 m s⁻¹ are significantly higher than those at 16 m s⁻¹ for all σ, in contrast to the trends of void fractions in the shear layer, which are ~4 times higher at 16 m s⁻¹. This trend is likely related to differences in the absolute mean pressure in the facility, keeping in mind that for the same σ, e.g. σ = 0.5, the pressure at 10.5 m s⁻¹ is 0.46 bar, and that at 16 m s⁻¹ is 1.04 bar. Consequently, while the dissolved oxygen level is nearly the same in both cases (5.5 ppm), the saturation levels of non-condensable gases are different, 130 % at 10.5 m s⁻¹, and 75 % at 16 m s⁻¹. Hence, the water in the test section is supersaturated at 10.5 m s⁻¹ and undersaturated at 16 m s⁻¹. As a result, in the relatively quiescent region in the recirculating zone (0.1U), mass diffusion is expected to increase the number density of detected bubbles at 10.5 m s⁻¹ and decrease it at 16 m s⁻¹. Since some of these bubbles are likely to be entrained into the shear layer, becoming nucleation sites there, one should not be surprised to find higher concentration of small nuclei (<200 μm) at the lower velocity in the shear layer as well. Conversely, the concentration of large bubble at 16 m s⁻¹ exceeds that at 10.5 m s⁻¹, presumably owing to differences in generation by local cavitation events (figure 5b). Indeed, at the high end, the size of these bubbles is comparable to that of the cavities. The slopes of the number density distributions also increase from −5 to −2 with decreasing cavitation index. In summary, the small nuclei concentration in the shear layer is higher at 10.5 m s⁻¹ than that at 16 m s⁻¹, in clear contrast to the void fraction of cavitation events. Hence, availability of nuclei is not the reason for the increase in the number of cavitation events with increasing velocity. These observations support the arguments made in the previous section that the spatial distribution of cavitation events is dominated by the structure of the pressure field, namely the size, shape, location and duration of low pressure events inside the QSVs.

Figure 28. (a) The location of ‘natural’ bubble distribution measurements shown on the spatial distribution of cavities at 16 m s⁻¹ and σ = 0.55. (b–d) The nuclei number densities for the indicated velocities and cavitation indices, all slightly below cavitation inception levels, in the: (b) free stream, (c) recirculation region under the shear layer and (d) shear layer. Dotted lines show the indicated slopes.

To determine how the nuclei distributions are affected by entrainment of free-stream bubbles into the low pressure regions of the shear layer, we compare the spatial and size distributions of bubbles under controlled seeding at a pressure high enough to prevent cavitation behind the step. We intentionally perform experiments at the same pressure to minimize bias caused by mass diffusion, as discussed above. The nuclei number densities at three different speeds (1.45, 10.5 and 16 m s⁻¹) are shown in figure 29(a). Here, the free-stream and shear layer data are presented in the same plot to allow direct comparisons. Note that the size range is quite limited and does not differ significantly from the injection diameter of 60 μm. In general, there are small difference in the free-stream (FS) size distributions with the bubbles at 10.5 m s⁻¹ being slightly larger than those of the other speeds, yet the overall concentrations differ by less than 11 %. The free-stream concentrations do not vary significantly with axial and spanwise location (not shown). Within the shear layer there is minimal difference between the number densities at 10.5 and 16 m s⁻¹, but the concentration at 1.45 m s⁻¹ is smaller in the 50–70 μm size range, presumably owing to weaker entrainment into low pressure regions within eddies.

Figure 29. Data for controlled bubble injection at a mean tunnel pressure of 1.1 bar: (a) nuclei number densities in the free stream, denoted as FS, and in the shear layer (SL) for the indicated velocities. The sample areas are marked in the inset. (b) The corresponding vertical profiles of microbubble concentration.

Vertical profiles of bubble concentrations in the shear layer and the free stream are provided in figure 29(b), this time using a linear scale to highlight differences. The error bars are based on a bootstrap analysis of 5 subsamples. In all cases, the profiles have two maxima, with the larger one located at approximately y/h = 1.2, well above the shear layer. The free-stream concentration is consistent with the bubble injection rate, and by design, the magnitude and location of the bubble concentration peak show minimal effects of velocity. It appears that significant fractions of the injected bubbles end up in the free stream, and do not play any role in cavitation inception. The combined effects of buoyant rise (1.8 mm s⁻¹ for a 60 μm bubble) and vertical pressure gradients, which oppose each other, are not significant to be observed in the free-stream profile. The second and slightly lower peak is located inside the shear layer, in the region of high mean velocity gradients (figure 3), peak Reynolds shear stresses (figure 4) and the same area where cavitation inception is most prevalent. All the concentrations are lower in the shear layer, 35 % of the free-stream values at 1.45 m s⁻¹ and 75 % at the higher speeds, but the characteristic sizes of bubbles are slightly larger. Identification of possible reasons for this trend, e.g. selective entrainment of large bubbles, turbulence-induced rectified diffusion and even residence time in the recirculating region below the shear layer, is beyond the present scope. Note that, for a flow field with similar characteristics, i.e. the same scaled velocity and pressure fields, the actual pressure gradients induced by the shear layer eddies and, therefore, the associated entrainment rates, increase with velocity. At the higher velocities, the maximum concentrations do not differ significantly from each other, and they are both approximately 75 % of the free-stream values. Hence, they do not exhibit trends that would explain the differences in the number of cavitation events. Between the two peaks, especially in the upper edge of the shear layer at this axial location, 0.0 < y/h < 0.5 (figure 4d), the bubble concentration is very low, suggesting that they might have been depleted by entrainment into the middle of the shear layer. It should be noted that Allan et al. (Reference Allan, Barbaca, Russell, Venning, Pearce and Brandner2022) also report an inward entrainment of bubbles, which are injected into the boundary layer upstream of a downstream-facing step, at x/h > 3. They also observe that the nuclei concentration, above a certain threshold, does not seem to have a major effect on the rate of cavitation events.

Other effects can be inferred from the statistics of nuclei distributions and motions, as summarized in figure 30. First, figure 30(a) shows several sample bubble trajectories at 16 m s⁻¹. Close to the free stream, at the upper edge of the shear layer, the bubble trajectories are more likely to be nearly straight. In contrast, deeper in the shear layer, the bubble trajectories become complex, and they appear to slow down and linger for a brief period of approximately 0.1h/U before accelerating again. The trajectory segments where the bubbles are ‘trapped’, i.e. where their speed is lower than 0.06U, are marked in red. This selected threshold is approximately 50 % of the r.m.s. of the liquid vertical velocity fluctuations, 0.12U, based on the 2-D PIV data in figure 4. As is evident, the trapped segments frequently have small radii of curvature, a phenomenon that can be attributed to interactions with the shear layer vortices. Such trapping has been seen at all velocities for approximately 10 % of the nuclei. Characteristic kinked trajectories have also been observed for bubbles entrained by vortex rings (Sridhar & Katz Reference Sridhar and Katz1999). The vertical profiles on the mean bubble streamwise velocity are compared with the liquid velocity in the shear layer in figure 30(b). In the middle of the shear layer, i.e. at −0.5 < y/h < −0.1, where the bubble concentrations and the Reynolds stresses peak, the bubbles are slower than the mean flow at all speeds. Above and below this region, the differences from the mean liquid velocity diminish. The shape and location of the bubble trajectories suggest that the observed trapping is at least one of the mechanisms for the reduced bubble speed. Trends with velocity are difficult to decipher, and do not deviate beyond the uncertainty determined based on bootstrapping. In summary, the sample results in figures 29 and 30 suggest that interactions between bubbles and vortices affect the bubble dynamics to a level that cannot be ignored in assessing the conditions for cavitation inception. There is a considerable increase in entrainment of bubbles into the shear layer as the velocity increases from 1.45 to 10.5 or 16 m s⁻¹. Yet, the concentrations, as well as spatial and size distributions at 10.5 and 16 m s⁻¹ are quite similar. Hence, they do not point at nuclei-related dynamics that affects the observed rates of cavitation events.

Figure 30. (a) Sample trajectories at U = 16 m s⁻¹ containing ‘trapped’ bubbles highlighted in red; and (b) a comparison of the vertical profiles of bubble mean streamwise velocity with that of the liquid. Error bars indicate the standard deviation.