3.1. Tip leakage cavitation

In this section, we discuss the location of cavitation inception, the magnitude of cavitation inception indices, and the evolution of cavitation with decreasing pressure as a function of advance ratio. Sample images of cavitation in the tip region for different cavitation indices ( $\sigma$ ), all at design J (=0.85), are shown in figure 5. Here, $\sigma =(p_{in}-p_{v})/0.5\rho V_{z}^{2}$ , where p _in is the centre line pressure at the inlet to the test section, p _v is the vapour pressure of the NaI solution at 20-25°C (=1.2 kPa, Patil et al., Reference Patil, Tripathi, Pathak and Katti1991) and V _z = 2.3 ms⁻¹. A white line is drawn along the blade tip to highlight its location, and the insets presented in the second row are enlarged views of the encased areas. As demonstrated by supplementary movie 1 and figure 5(a), at J = 0.85, cavitation inception typically appears as almost axially aligned, 1-2 mm long, elongated bubbles that are located circumferentially downstream and axially upstream of the blade tip TE. Based on their nearly axial orientation (figure 5(a, d–f)), these inception events appear to occur in secondary structures and last less than 1 millisecond (figure 5(d–f)), consistent with observations reported in Chesnakas and Jessup (Reference Chesnakas and Jessup2003) for a different propeller. A reduction in $\sigma$ extends the cavitation to the primary TLV in the region (figure 5(b)). Further reduction in pressure extends the cavitation to the entire TLV, and then to travelling bubble cavitation within the blade tip gap, and between the SS of the blade tip and the TLV (figure 5(c)). Figure 5(c) can also be used for visualising the TLV centre, showing that it rolls up at mid-chord and then migrates away from the blade SS while being aligned in the circumferential direction. The corresponding video, supplementary movie 2, also demonstrates the presence of multiple secondary vortices (presumably) that begin to cavitate intermittently only in the vicinity of the TLV. These vortices are located radially above the TLV and oriented perpendicularly to it before wrapping helically around the TLV centre. Collectively, figure 5 and the movies confirm that cavitation inception indeed occurs in the nearly axial secondary structures. Furthermore, as the secondary structures cavitate only in the vicinity of the TLV, it appears that they are subjected to TLV-induced stretching, which reduces the pressure in their core. A similar evolution of incipient phenomena is observed at J = 0.933 (not shown), except for differences in the location of TLV rollup and trajectory, as demonstrated later.

Figure 5. Sample examples of cavitation for J = 0.85: (a) σ = 10.6, (b) σ = 8.3, (c) σ = 6.5. Corresponding magnified views of the regions highlighted by blue boxes are shown in the bottom row: (d–f) time evolution of the inception bubble at σ = 10.6, (g) magnified view at σ = 8.3, (h) magnified view at σ = 6.5.

Figure 6. Sample examples of cavitation for J = 0.68: (a) σ = 33.8, (b) σ = 20.6, (c) σ = 14.8, (d) σ = 12.2, (e) σ = 10. A magnified view of the region highlighted with a blue box in (a) is shown on the left in the bottom row.

Figure 6 shows the evolution of cavitation at J = 0.68, an off-design case, with decreasing $\sigma$ . The inception bubble (figure 6(a), and its enlarged view in the figure inset) forms in the mid-chord region and appears to be oriented along the TLV axis, circumferentially and axially upstream of the blade tip TE. A reduction in pressure expands the cavitation, first to part (figure 6(b)), and then to the entire TLV trajectory and secondary vortices entrained by it (figure 6(c)). Compared with design conditions, at this J, the TLV rollup begins at a lower s/c, and the secondary structures appear circumferentially upstream as well as downstream of the TE. Travelling bubble cavitation in the tip gap, expansion of the cavitating secondary vortices and filling of the space between the SS tip and the TLV start to occur at $\sigma$ = 12.2 (figure 6(d)). At this stage, the cavitating TLV is located farther from the blade SS, indicating that cavitation also alters the flow field in the tip region. Further reduction in pressure alters the entire cavitation structure, with the cavitating TLV disappearing and being replaced by multiple large and violent vortices extending over a broad area (figure 6(e)). Figure 7 shows the cavitation inception indices for different J, scaled using V _z ( $\sigma _{i}$ , figure 7(a), and U _T ( $\sigma _{i,{U_{T}}}$ , figure 7(b)). The displayed values correspond to the pressure below which cavitation is first observed at least once every two rotations of the machine. The error bars account for the uncertainties in pressure, velocity, as well as the standard deviation among multiple tests involving both blades. With decreasing J, inception occurs at a higher pressure, resulting in higher $\sigma _{i}$ . However, when scaled with the varying U _T , $\sigma _{i,{U_{T}}}$ flattens at about 0.95 for all the data obtained at J ≥ 0.80, within the uncertainty level. The current values of $\sigma _{i}$ are 40%, 17% and 7% lower than those reported by Michael et al. (Reference Michael, Choi and Otero2024) for J = 0.80, 0.85 and 0.93, respectively, at an order of magnitude higher Re.

Figure 7. Cavitation inception index for varying J: (a) $\sigma _{i}$ scaled using V_z ; (b) $\sigma _{i,{U_{T}}}$ scaled using U_T .

3.2. Ensemble-averaged flow structure at design advance ratio

The section describes the mean flow structure at J = 0.85, as deduced from the phase-locked and the high-speed SPIV datasets, focusing on the origin and evolution of prominent flow features. Distributions of the ensemble-averaged circumferential vorticity, $\langle \omega _{\theta }\rangle /\Omega$ , where $\omega _{\theta }=(\partial u_{r}/\partial z-\partial u_{z}/\partial r)$ , in nine meridional planes (figure 3(b)), calculated using the higher resolution phase-locked SPIV data, are presented in figure 8. Note the difference in the axial extents of the FOV between the left and right columns. This figure also contains the in-plane ensemble-averaged velocity vectors (U _r , U _z ), with their spacing diluted by 2:1 in r and 3:1 in z for clarity. Plots of the ensemble-averaged circumferential velocity (U $_{\theta}$ /U _T ) at s/c = 0.75, 1.00 and 1.13, calculated from the same dataset, are presented in figure 9. At the LE (s/c = 0.00, figure 8(a)), the flow within the tip gap is still from the SS to the PS, whereas at s/c ≥ 0.25, as tip leakage begins, it is directed from the PS to the SS. Rollup of a TLV begins at around mid-chord (s/c = 0.50, figure 8(c)), with the appearance of a positive vorticity peak and flow swirling near the SS tip. At 0.75 ≤ s/c ≤ 1.00, the TLV grows and migrates away from the SS, while remaining ‘connected’ to the blade tip by a shear layer that defines the lower boundary of the backward leakage flow. Throughout its trajectory, the TLV centre is located in regions of both elevated magnitude and radial gradients of U $_{\theta}$ (figure 9(a–c)), indicating that this vortex is a 3-D swirling jet. Similar phenomena have been seen for TLVs of axial waterjet pumps (Wu et al., Reference Wu, Miorini and Katz2011a) and aviation compressors (Li et al., Reference Li, Chen, Tan and Katz2019; Saraswat et al., Reference Saraswat, Koley, Joly and Katz2025). The mechanisms affecting the TLV migration, which include flow induced by the leakage jet during early phases, and by the ‘image vortex’ on the other side of the endwall in the aft part of the rotor, are characterised in detail by Li et al. (Reference Li, Chen, Tan and Katz2019).

Figure 8. Distribution of $\langle \omega _{\theta }\rangle /\varOmega$ in nine meridional planes for J = 0.85, calculated using the phase-locked dataset. The superimposed in-plane velocity vectors (U _r , U _z ) are shown with their spacing diluted 2:1 in r and 3:1 in z for clarity.

Figure 9. Distribution of $U_{\theta }/U_{T}$ for J = 0.85 at (a) s/c = 0.75, (b) s/c = 1.00 and (c) s/c = 1.13. The white dotted lines in (b) highlight the velocity variations within the wake. The black lines in (a–c) are of $\langle \omega _{\theta }\rangle /\varOmega =40$ .

Starting from s/c = 0.50 (figure 8 c), impingement of the backflow on the forward passage flow, above and slightly upstream of the TLV, causes boundary layer separation on the duct wall. This phenomenon results in entrainment of positive vorticity originating from the forward flow boundary layer and negative vorticity from the leakage boundary layer, away from the wall. Driven in part by the TLV-induced flow, the negative vorticity layer is entrained away from the endwall and rolls up into a distinct counter-rotating vortex (CRV), marked with a red arrow in figure 8(f). Similar phenomena have been seen in all the ducted axial turbomachines that we have tested, including the above-mentioned waterjet pump and aviation compressors. The locations and strengths of the resulting vortex pairs (TLV and CRV) vary, depending on the blade loading, operating condition, tip gap size, etc.

In the forward part of the blade, $\langle \omega _{\theta }\rangle$ < 0 and $\langle \omega _{\theta }\rangle$ > 0 regions form on the blade SS and PS tips, respectively (figure 8(a,b)). When the TLV is located near the blade SS, shortly after rolling up, the induced radially upward flow below the SS tip corner further augments the negative $\langle \omega _{\theta }\rangle$ there (figure 8(d–e)). However, the effect of TLV diminishes as it migrates away from the blade (e.g., figure 8f). In the aft part of the blade (0.75 ≤ s/c ≤ 1.00, figure 8(d–f)), elevated positive and negative $\langle \omega _{\theta }\rangle$ layers appear along the blade SS and PS surfaces, respectively, and increase in magnitude close to the TE and in the near wake (figure 8(f)). These vortical layers have signs that are opposite to those at the blade tips (figure 8(d–e)). At s/c = 1.00 (figure 8(f)), where only the blade tip is visible, the SS contains two opposite-sign vorticity layers. However, circumferentially downstream of the blade (at s/c = 1.13, figure 8(g)), the negative $\langle \omega _{\theta }\rangle$ layers originating from the SS and PS merge and become a single continuous layer. The shear layer, located under the backward flow, in figure 8(g) also contains two additional secondary vortices (marked with black arrows), that are referred to as SV1 (on the left) and SV2 (on the right) in subsequent discussions. At higher s/c, these vortices are entrained into the TLV (e.g., figure 8(h)). By s/c = 1.50 (figure 8(i)), the distinct secondary vortices and the positive vorticity layer associated with the wake and the shear layer are all entrained into the TLV. The remaining broad region with negative vorticity along the upstream periphery of the TLV contains the CRV remnants.

Figure 10 shows sample distributions of all three ensemble-averaged vorticity components, obtained from the lower resolution high-speed SPIV data, in three selected planes, one before (s/c = 0.98) and the other two after (s/c = 1.03, 1.08) the blade TE. Here, the distributions of $\langle \omega _{r}\rangle /\Omega$ , where $\omega _{r}=({r^{-1}}\partial u_{z}/\partial \theta -\partial u_{\theta }/\partial z)$ , and $\langle \omega _{z}\rangle /\Omega$ , where $\omega _{z}=(\partial u_{\theta }/\partial r-{r^{-1}}\partial u_{r}/\partial \theta +u_{\theta }/r)$ , are plotted using the same colour scale as $\langle \omega _{\theta }\rangle /\Omega$ . To facilitate comparisons between components, all the plots also show line contours of specified $\langle \omega _{\theta }\rangle /{\Omega}$ . For this dataset, the averaging is based on only 114 realisations, hence, the distributions are more ‘jittery’. There are also slight differences in the locations of vortices between the two datasets, e.g., an axial shift of 0.03c _a in the location of the TLV centre at s/c = 1.00, which can be attributed to dissimilar spatial resolutions, interframe times, number of realisations and errors in flow rate. The sample snapshots demonstrate the rapid changes to the flow and vorticity fields near the TE. This evolution is further illustrated in supplementary movies 3-5, which show the mean radial, circumferential, and axial vorticity components, and supplementary movie 6 that displays the mean circumferential velocity. This set enables us to examine the evolution of the 3-D vorticity field.

Figure 10. Distribution of ensemble-averaged vorticity components for J = 0.85 calculated from the high-speed dataset at s/c = 0.98 (left), s/c = 1.03 (middle), s/c = 1.08 (right): (a, d, g) $\langle \omega _{r}\rangle /\varOmega$ , (b, e, h) $\langle \omega _{\theta }\rangle /\varOmega$ , (c, f, i) $\langle \omega _{z}\rangle /\varOmega$ . Line contours in all panels are of $\langle \omega _{\theta }\rangle /\varOmega$ marked in the middle row.

Starting with the blade surfaces, the pressure gradients between the PS and SS blade tips drive the flow radially up near the PS tip, and then backward across the tip gap. This process generates positive $\langle \omega _{\theta }\rangle$ on the PS tip corner, and on the blade tip surface (figure 10(b)), which becomes the source of the vorticity in the SS tip shear layer and rolls up into the TLV. The blade tip also has layers of $\langle \omega _{z}\rangle$ < 0 along the SS and the underside of the shear layer, and regions of $\langle \omega _{z}\rangle$ > 0 along the PS and the topside of the shear layer (figure 10(c)). The latter is associated with radial gradients of U $_{\theta}$ in the boundary layer along the tip gap. Near the TE, this component is comparable in magnitude to $\langle \omega _{\theta }\rangle$ . The negative $\langle \omega _{z}\rangle$ and the dominant positive $\langle \omega _{r}\rangle$ layers along the SS surface, and the $\langle \omega _{r}\rangle$ < 0 layer along the PS (figure 10(a)) are all associated with the blade boundary layer. The primary contributor to the radial vorticity is ∂U $_{\theta}$ /∂z (e.g., figure 9(b)). However, due to the slope/tilt of the blade surfaces in the meridional view, part of the boundary layer vorticity is aligned in the axial direction, contributing to $\langle \omega _{z}\rangle$ < 0 in the SS and $\langle \omega _{z}\rangle$ > 0 on the PS. In addition, inherently ∂U $_{\theta}$ /∂r < 0 along the surface of any blade rotating in the negative $\theta$ direction, contributing to $\langle \omega _{z}\rangle$ < 0 on both sides. The combined effects of these two mechanisms result in a slightly positive $\langle \omega _{z}\rangle$ on the PS, and a negative $\langle \omega _{z}\rangle$ , with a higher magnitude, on the SS. Finally, $\langle \omega _{\theta }\rangle$ is the smallest near-surface component, but still has positive and negative values on the SS and PS surfaces, respectively (figure 10(b)), with the PS values being an order of magnitude lower than those on the SS. The origin of this component is elucidated later.

Just behind the blade, at s/c = 1.03 (figure 10(d–f)), the 3-D vorticity layers originating from the blade SS and PS surfaces persist in the near wake. The magnitude of $\langle \omega _{r}\rangle$ remains of the same order as that in the blade boundary layer, but $\langle \omega _{\theta }\rangle$ increases significantly. As for the vorticity originating from the tip, the positive $\langle \omega _{z}\rangle$ layer is still evident (figure 10(f)). Further behind the blade (s/c = 1.08, figures 10 g-i), all three components of the wake vorticity diffuse, especially the SS wake, and the $\langle \omega _{z}\rangle \gt 0$ layer originating from the blade tip gap is advected radially upwards, eventually disappearing at higher s/c. Subsequently (see supplementary movie 5), the distribution of $\langle \omega _{z}\rangle$ is dominated by the negative layer entrained from the SS side of the blade wake and, to a lesser extent (shown later), tilting of the radial vorticity originated from the SS surface (figure 10(d)). The distributions of $\langle \omega _{\theta }\rangle$ at s/c = 1.03 and 1.08 show the presence of SV1 and SV2 (figure 10(e, h)), that co-rotate with the TLV. To understand their origin, figure 11 (see also supplementary movie 4) shows ensemble-averaged distributions of $\langle \omega _{\theta }\rangle /{\Omega}$ in closely spaced planes, focusing on the vicinity of the TE. The origin of SV1 can be traced back to the shear layer separating the backward leakage flow from the forward passage flow radially inward from it (figure 11(d)). The source of this vorticity is the blade PS tip corner. In contrast, SV2 starts to roll up circumferentially downstream of the blade, appearing first in the present results at s/c = 1.02, and its vorticity appears to largely originate from the blade SS wake (figure 11(f)). The distributions of U $_{\theta}$ (supplementary movie 6, or figure 9 b-c) indicate that both SV1 and SV2 are located in a flow with elevated negative U $_{\theta}$ , which is entrained from the near wake as well as the blade PS upstream of the TE. Hence, describing them as a pair of secondary circumferential vortices underestimates the flow complexity. In reality, similar to the TLV, both SV1 and SV2 are swirling jets.

Figure 11. Distributions of $\langle \omega _{\theta }\rangle /\varOmega$ in closely spaced s/c locations near the blade TE tip for J = 0.85, calculated using the high-speed dataset.

This paragraph discusses the origin of the $\langle \omega _{\theta }\rangle$ layers on the blade surface (figure 10(b)) that contribute to the formation of SV2. Owing to the no-slip condition, $(1/\rho )\partial p/\partial r\approx {u_{\theta }}^{2}/r+\nu \partial \omega _{\theta }/\partial z$ on the blade surfaces. The second term on the right-hand side represents wall-normal viscous diffusion of vorticity. Since the present geometry is tip loaded by design (Michael et al., Reference Michael, Choi and Otero2024), $\partial p/\partial r$ is negative over a substantial fraction of the blade surfaces, especially the SS (except for the vicinity of the TE), resulting in $\partial \omega _{\theta }/\partial z\lt 0$ on both sides. Hence, viscous diffusion of vorticity away from the blade surface should add $\langle \omega _{\theta }\rangle \gt 0$ to the PS and $\langle \omega _{\theta }\rangle \lt 0$ to the SS, both in contrast to the present observations in the aft part of the blade (e.g. figure 10(b)). Presumably, either the high-speed SPIV resolution is not sufficient for resolving the inner part of the boundary layers, or other phenomena are stronger. On the PS, the broad low magnitude $\langle \omega _{\theta }\rangle \lt 0$ layer starts from the forward part of the blade (figure 8(b, c)). The primary term contributing to this vorticity, ∂U _r /∂z, is negative outside of the PS boundary layer. In this region, there is radially outward flow, which diminishes away from the blade, as demonstrated by the velocity vectors in figure 8(c–e). This outward flow is entrained into the tip gap, suggesting that it is pressure driven. The origin of the negative vorticity is located in the forward part or upstream of the blade. Considering that similar layers appear in other axial turbomachines with varying load distributions (Li et al., Reference Li, Chen, Tan and Katz2019), the mechanism involved apparently occurs in all of them. One of the potential sources is the axial flow in the hub boundary layer, where $\langle \omega _{\theta }\rangle \lt 0$ . Shifting to the SS, the origin of the $\langle \omega _{\theta }\rangle \gt 0$ layer involves the tilting of other components by the mean flow, such as the reorientation of $\langle \omega _{z}\rangle$ associated with $\langle \omega _{z}\rangle \partial U_{\theta }/\partial z$ , or tiling of $\langle \omega _{r}\rangle$ , caused by $\langle \omega _{r}\rangle \partial U_{\theta }/\partial r$ . Figure 12 shows the distribution of the relevant terms at several s/c. In the aft part of the blade SS, e.g., at s/c = 0.92, the two tilting terms have opposite signs, but their sum is positive (figure 12(a)). In addition, figure 12(b) shows that $\langle \omega _{\theta }\rangle$ is subjected to circumferential stretching ( $\langle \omega _{\theta }\rangle r^{-1}\partial U_{\theta }/\partial \theta$ > 0) by the mean flow on the SS. Being much higher than the tilting terms, such stretching should be a major contributor to the increase in $\langle \omega _{\theta }\rangle$ near the TE. Finally, there are two possible sources for the $\langle \omega _{\theta }\rangle \lt 0$ layer that increase in thickness near the SS TE. The first is viscous diffusion associated with the radial pressure gradients, as discussed above. The second is the radial outward flow induced by the TLV near the tip corner, which occurs only after the vortex rolls up (figure 8(d–e)). Consequently, the SS TE contains two vorticity layers with opposite signs. Such multi-layered SS vorticity has been observed experimentally by Chesnakas and Jessup (Reference Chesnakas and Jessup2003), and computationally by Leasca et al. (Reference Leasca, Kroll and Mahesh2024) in another propeller, i.e., it is not unique to the present geometry.

Figure 12. Distribution of tilting, stretching and advection of vorticity components by the mean flow for J = 0.85 at (a, b) s/c = 0.92, (c, d) s/c = 1.00, (e–j) s/c = 1.03. Definitions of each term are provided on the individual panels. The line contours of $\langle \omega _{\theta }\rangle /\varOmega$ at s/c = 1.03 match those in figure 10(d–f). Colour scale for panels (a–i) is provided on top, and as a separate inset for (j).

As mentioned before, in the near wake, the $\langle \omega _{\theta }\rangle \lt 0$ layers from both sides of the blade merge, leaving only two distinct layers with opposite signs (e.g. figure 11(f)). Strong circumferential stretching of the $\langle \omega _{\theta }\rangle$ > 0 layer (figure 12(d, f)) persists up to the region where SV2 first appears, contributing to the increase in $\langle \omega _{\theta }\rangle$ , and overcoming the opposing effects of tilting (figure 12(e, g–h)). The circumferentially aligned SV1 and SV2 partially overlap with the perpendicular negative $\langle \omega _{z}\rangle$ layer (figure 10(h, i)). The latter originates mostly from the blade SS (figure 10(c)), but as demonstrated in figure 12(i), it is also affected by tilting of the positive $\langle \omega _{r}\rangle$ layer generated in the blade SS boundary layer. However, the peaks of $\langle \omega _{\theta }\rangle$ and $\langle \omega _{z}\rangle$ in these secondary vortices do not coincide, with the latter occurring radially inward from the former. This lattice of vortices is of particular importance since, as demonstrated later, cavitation inception preferentially occurs in the negative $\langle \omega _{z}\rangle$ layer, especially in the region where the axial vorticity is stretched axially by the flow induced by the TLV as well as by SV1 and SV2. Before proceeding, it should be noted that figures 10 and 12, and supplementary movies 3–5 show numerous other phenomena affecting the evolution of the TLV, such as strain-induced reorientations of $\langle \omega _{r}\rangle$ and $\langle \omega _{z}\rangle$ layers, entrainment and merging of vortices, and non-uniform advection by the mean flow (e.g., figure 12(j)). Several of these interactions are described in Wu et al. (Reference Wu, Tan, Miorini and Katz2011 b) and Li et al. (Reference Li, Chen, Tan and Katz2019) for other axial turbomachines. While they also occur in the present propeller, they are not repeated here to maintain brevity, as we restrict ourselves to phenomena affecting cavitation inception.

3.3. Comparison between mean flow structure and cavitation inception at design advance ratio

Aimed at identifying the physical mechanisms governing the onset of cavitation, this section compares the measured flow field to the observed location of cavitation inception at J = 0.85. By combining results obtained in the closely spaced meridional planes of the high-speed dataset, one can reconstruct the flow structure along radial cuts (constant r ^*) as illustrated in figure 13. Sample cuts selected for demonstrating trends are r ^* = 0.985, which is located radially above the blade tip, as well as r ^* = 0.974, 0.969 and 0.959, which intersect with the blade tip. For reference, the blade tip extends to r ^* = 0.979. Figure 14 shows the distributions of $\langle \omega _{\theta }\rangle$ and $\langle \omega _{z}\rangle$ in the near-TE region of the selected radial cuts. Above the blade tip, figure 14(a) shows a broad area with elevated $\langle \omega _{\theta }\rangle$ axially upstream of the SS, which corresponds to the shear layer connecting the SS tip to the TLV (e.g. figure 10(b)). This layer also contains two diagonal bands of particularly high $\langle \omega _{\theta }\rangle$ that corresponds to SV1 and SV2. Both start near the TE and extend axially upstream and circumferentially downstream of it. The shear layer vorticity peaks at r ^* = 0.974 (figure 14(c)) and decreases further inwards. At r ^* = 0.959, the signatures of SV1 and SV2 diminish, leaving the TLV as the most prominent structure (figure 14(g)). The negative $\langle \omega _{\theta }\rangle$ layer axially upstream of the TLV, which persists in all the distributions, corresponds to the signature of the separating duct boundary layer and rollup of the CRV. Prominent features in the distribution of $\langle \omega _{z}\rangle$ include: (i) the positive region associated with the blade tip (figure 14(b)) and PS (figure 14(b, d, f, h)) boundary layers, which extends to the near wake, (ii) the broad negative region originating from the blade SS that persists in lower radial cuts (figure 14(f, h)), (iii) the previously discussed negative vorticity radially inward from SV1 and SV2 (figure 14(d, f)), which is advected radially outwards behind the blade (figure 14(b)), and (iv) the positive layer along the TLV, which is associated with the alignment of its trajectory (figure 14(h)).

Figure 13. Illustration of reconstructed radial cuts (constant r^*) from closely spaced meridional planes in the high-speed dataset. Locations of four specific radial cuts are indicated.

Figure 14. Distributions of ensemble-averaged vorticity components for J = 0.85 in reconstructed radial cuts: (a, c, e, g) $\langle \omega _{\theta }\rangle /\varOmega$ , (b, d, f, h) $\langle \omega _{z}\rangle /\varOmega$ . Panels show (a, b) r^* = 0.985, (c, d) r^* = 0.974, (e, f) r^* = 0.969, (g, h) r^* = 0.959. The black lines are of $\langle \omega _{\theta }\rangle /\varOmega$ =40. The white markers denote the s/c and z/c_a locations of inception events. An outline of the blade tip is overlaid on panels (a) and (b).

All the plots in figure 14 also show white markers indicating the measured locations of cavitation inception events. We have not observed differences between the dataset obtained near the two blades; hence both are included together. As mentioned earlier, while the radial locations of these events cannot be ascertained based on the image recorded by a single camera (figure 2(a)), supplementary movie 2 indicates, based on how the cavitating secondary structures wrap around the TLV, that they are located radially outward of the TLV centre. Most of the events appear to be clustered in two different regions. The first group, subsequently referred to as CI₁, is mostly located at 0.98 ≤ s/c ≤ 1.1 and 0.70 ≤ z/c _a ≤ 0.80, and the second (CI₂), at 1.09 ≤ s/c ≤ 1.16 and 0.75 ≤ z/c _a ≤0.85. These ranges agree with those reported for the larger propeller in Michael et al. (Reference Michael, Choi and Otero2024). The left column of figure 14 clearly indicates that these regions do not coincide with the peaks of $\langle \omega _{\theta }\rangle$ associated with SV1 and SV2. They partially overlap with the region of elevated negative $\langle \omega _{z}\rangle$ (figure 14, right column), but also do not coincide with the regions of peak vorticity magnitude. In contrast, the right column of figure 15 shows that both CI₁ and CI₂ appear to be concentrated in regions with high magnitude of axial stretching of $\langle \omega _{z}\rangle$ by the mean flow, i.e. $\langle \omega _{z}\rangle \partial U_{z}/\partial z,$ but generally do not agree with the location of elevated circumferential stretching of $\langle \omega _{\theta }\rangle$ , i.e., $\langle \omega _{\theta }\rangle r^{-1}\partial U_{\theta }/\partial \theta$ , shown in the left column. Although, away from the blade, the magnitude of $\langle \omega _{\theta }\rangle$ is larger than that of $\langle \omega _{z}\rangle$ (figure 14), $\partial U_{z}/\partial z$ is significantly larger than $r^{-1}\partial U_{\theta }/\partial \theta$ , especially in the cavitation inception region. More specifically, group CI₂ appears to be co-located better with the stretching peak at r ^* = 0.985 (figure 15(b)), i.e., above the blade tip, and CI₁ is better co-located with the stretching peak at r ^* = 0.974 and 0.969 (figure 15(d, f)).

Figure 15. Distributions of vorticity stretching by mean flow terms for J = 0.85 in reconstructed radial cuts: (a, c, e, g) $(\langle \omega _{\theta }\rangle r^{-1}\partial U_{\theta }/\partial \theta )/\varOmega ^{2}$ , and (b, d, f, h) $(\langle \omega _{z}\rangle \partial U_{z}/\partial z)/\varOmega ^{2}$ . Panels show (a, b) r^* = 0.985, (c, d) r^* = 0.974, (e, f) r^* = 0.969, (g, h) r^* = 0.959. Colour contours are 3 × 3 median-filtered for display. The black lines are of $\langle \omega _{\theta }\rangle /\varOmega$ =40. White markers denote the s/c and z/c_a locations of inception events. An outline of the blade tip is overlaid on panels (a) and (b).

Further clarification of the mechanisms involved is provided in figure 16. This plot highlights $\langle \omega _{z}\rangle \partial U_{z}/\partial z$ in only those regions where it and $\langle \omega _{z}\rangle$ are negative, i.e., where the negative axial vorticity is stretched axially, and compares them with contours of $\langle \omega _{\theta }\rangle$ , showing where the TLV, SV1 and SV2 are located, as well as to the area where the magnitude of negative $\langle \omega _{z}\rangle$ is high. The vertical lines in figure 16(a) indicate the axial boundaries of the CI₁ area at s/c = 1.05, and those in figure 16(b) bound the CI₂ region at s/c = 1.13. The four grey horizontal lines correspond to the r ^* of the radial cuts presented in figures 13, 14 and 15. As is evident, the peak vorticity stretching in the CI₁ area is located between SV1 and the TLV, and the axial stretching is caused by the opposite axial velocity induced by two vortices. At s/c = 1.13, SV1 is already being entrained into the TLV, and its remnants are located above the TLV centre, while SV2 is still distinct. The CI₂ region is now located between SV1 (or the TLV under it) and SV2. In this case, the stretching is induced by the opposite axial flow induced by SV2 on the right, and TLV + SV1 on the left. In summary, both CI₁ and CI₂ regions are stretched axially by multiple vortices that are largely aligned in the circumferential direction, but the structures involved are different. As both C1 and C2 regions also have significant magnitude of $\langle \omega _{\theta }\rangle$ , the vortices undergoing stretching are ‘quasi-axial’ in orientation. Finally, one should keep in mind that figures 14, 15 and 16 still show ensemble-averaged quantities. While one cannot obtain the instantaneous $\omega _{z}$ from the present results, it is still possible to examine the axial straining in the cavitation inception regions. Figure 17 presents sample distributions of 3 × 3 median-filtered instantaneous $\omega_{\theta}$ with superimposed contours of $\partial u_{z}/\partial z$ . The regions where SV1 and SV2 are located contain multiple (circumferential) vorticity peaks, which induce high axial straining in the CI₁ (figure 16(a)) and CI₂ (figure 16(b)) regions. Similar to the trends of vorticity, the instantaneous axial straining is 3–5 times larger than the ensemble-averaged values. These observations provide motivation for performing volumetric 3-D velocity measurements of straining, 3-D vorticity and pressure, as performed in the regions of cavitation inception, e.g., of a turbulent shear layer by Agarwal et al. (Reference Agarwal, Ram, Lu and Katz2023).

Figure 16. Distributions of $(\langle \omega _{z}\rangle \partial U_{z}/\partial z)/\varOmega ^{2}$ in regions where $\langle \omega _{z}\rangle$ < 0 and $\langle \omega _{z}\rangle \partial U_{z}/\partial z$ < 0 for J = 0.85: (a) s/c = 1.05, (b) s/c = 1.13. Line contours: magenta- $\langle \omega _{\theta }\rangle /\varOmega$ = 45, purple- $\langle \omega _{\theta }\rangle /\varOmega$ = 30, yellow- $\langle \omega _{z}\rangle /\varOmega$ = −20. Grey horizontal lines mark r^* = 0.985, 0.974, 0.969, 0.959. The black vertical lines bound the axial extent of CI₁ in (a) and CI₂ in (b). The superimposed in-plane velocity vectors (U_r, U_z) are diluted 2:1 in both r and z for clarity.

Figure 17. Samples distributions of 3 × 3 median-filtered instantaneous circumferential vorticity ( $\omega _{\theta }/\varOmega$ ) for J = 0.85: (a) s/c = 1.05, (b) s/c = 1.13. The black vertical lines bound the axial extent of CI₁ in (a) and CI₂ in (b). The solid and dashed black lines are of $(\partial u_{z}/\partial z)/\varOmega$ = 20 and 5, respectively.

3.4. Effect of advance ratio on the flow structure

This section describes the effect of varying operating conditions on the mean flow features and the resulting changes to the location of cavitation inception in the tip region. Figure 18 compares the distribution of $\langle \omega _{\theta }\rangle /{\Omega}$ at s/c = 0.98 in the left column, $\langle \omega _{\theta }\rangle /{\Omega}$ at s/c = 1.03 in the middle column and $\langle \omega _{z}\rangle /{\Omega}$ at s/c = 1.03 in the right column, for different J values. Figure 19 provides quantitative data on the evolution of several tip flow parameters, all calculated using the high-resolution phase-locked dataset. As is evident from figure 18, the strength and location of the TLV and the CRV vary significantly with J. The mechanisms affecting these trends, which occur to varying extents in all the ducted axial turbomachines that we have tested, are associated with differences in blade loading. Consistent with the computational results of Michael et al. (Reference Michael, Choi and Otero2024) and Leasca et al. (Reference Leasca, Kroll and Mahesh2024), the latter for a different propeller, with decreasing flow rate, (i) the magnitude of blade loading, hence the strength of the TLV, is expected to increase, and (ii) the point of peak loading, hence the location of TLV rollup, is expected to move closer to the LE. For the present data, at s/c < 1.00, calculations of the total positive circulation on the blade SS ( $\Gamma _{\textit{pos}}$ , figure 19(a)) include regions of positive $\langle \omega _{\theta }\rangle$ around the TLV and the shear layer, at 0.8 ≤ $r^{*}$ ≤ 1. At s/c ≥ 1.00, SV1, SV2 and the vorticity in the blade SS wake are also included as they are continuously entrained into the TLV. An appropriate threshold, ranging between 2.5% and 5% of the maximum vorticity, is applied to avoid regions associated with the duct boundary layer, as part of it is also entrained into the TLV (figure 8(d–e)). To obtain the TLV circulation $(\Gamma _{TLV}$ , figure 19(b)), the shear layer, SV1 and SV2 are not included. Prior to entrainment, these secondary vortices are identified using the local vorticity maxima in their core and the minima between them. Once entrained, these secondary vortices are considered part of the TLV. The expected increase in blade loading with decreasing J agrees with the present distributions of $\Gamma _{\textit{pos}}$ , $\Gamma _{TLV}$ and the peak $\langle \omega _{\theta }\rangle$ in the TLV (figure 19(c)).

Figure 18. Distributions of ensemble-averaged vorticity components calculated using the high-speed dataset: (left) – $\langle \omega _{\theta }\rangle /\varOmega$ at s/c = 0.98, (middle) – $\langle \omega _{\theta }\rangle /\varOmega$ at s/c = 1.03, (right) – $\langle \omega _{z}\rangle /\varOmega$ at s/c = 1.03. (a–c) J = 0.933, (d–f) J = 0.85, (g–i) J = 0.803 and (j–l) J = 0.68. Black lines denote the specified values of $\langle \omega _{\theta }\rangle /\varOmega$ .

Figure 19. Effect of advance ratio on the evolution of ensemble-averaged tip flow parameters and TLV location, determined from the phase-locked dataset: (a) total positive circulation shed from the blade SS, $\Gamma _{\textit{pos}}/U_{T}c$ ; (b) circulation within the TLV, $\Gamma _{TLV}/U_{T}c$ ; (c) maximum circumferential vorticity in the TLV, ( $\langle \omega _{\theta }\rangle /\varOmega)_{max,TLV}$ ; (d) axial location of the TLV centre, $(z/c_{a})_{TLV}$ ; (e) radial location of the TLV centre, $(r^{*})_{TLV}$ ; (f) radially averaged magnitude of axial velocity at the SS of the tip gap, $U_{z,leak}/U_{T}$ ; (g) radially averaged magnitude of circumferential velocity at the SS of the tip gap, $U_{\theta ,leak}/U_{T}$ ; (h) radially averaged magnitude of chordwise normal velocity component relative to the blade at the SS of the tip gap, $U_{n,leak}^{*}/U_{T}$ .

Figure 19(d) shows that as the TLV evolves, its centre, identified based on the peak of $\langle \omega _{\theta }\rangle$ , migrates away from the blade SS (see also figure 8) at a rate that increases with decreasing J. The point of backflow boundary layer separation also follows similar trends (not shown). To understand the effect of operating conditions, it is shown in Li et al. (Reference Li, Chen, Tan and Katz2019) for an axial compressor that the TLV migration shortly after rollup is dominated by the tip leakage jet, and during later phases, by flow induced by the TLV’s ‘image vortex’ on the other side of the endwall. The present extent of leakage is characterised by radially averaging the velocity across the SS of the tip gap. Figure 19(f–h) show the evolution of the axial and circumferential velocity components in the laboratory reference frame ( $U_{z,leak},\; U_{\theta ,leak}$ ), as well as the chord-normal velocity component relative to the blade ( $U_{n,leak}^{*}$ ), respectively. As is evident, except for the vicinity of the TE, the magnitude of $U_{z,leak}$ and $U_{n,leak}^{*}$ increase significantly with decreasing J, a trend consistent with the increase in blade loading. At J = 0.68, $U_{n,leak}^{*}$ reaches 80% of the U _T at mid-chord, considerably earlier than the ∼ 70% peak measured at higher J values. Hence, the earlier and faster leakage jet at the lowest J is expected to push the TLV farther away from the blade. Furthermore, the corresponding higher TLV strength implies that the flow induced by its image is also stronger, affecting the TLV migration in later stages. These observations are consistent with the trends depicted in the left column of figure 18. Figure 19 shows several other interesting phenomena. First, along the blade, the TLV centre migrates further radially inward with decreasing J (figure 19(e)). As discussed in Li et al. (Reference Li, Chen, Tan and Katz2019), the likely causes for this trend include flow induced by the negative CRV originating from the separated endwall boundary layer, growth of the TLV, and flow induced by the secondary vortices in the shear layer. The only exception is the trend at J = 0.68, where the centre turns radially upwards at 1.0 ≤ s/c ≤ 1.5. As supplementary movie 9 and figure 18(k) demonstrate, this motion occurs while entrained secondary vortices and the shear layer wrap around the TLV, and are located axially upstream of its centre. Hence, the upward motion might be induced by secondary vortices. Once they are entrained, at s/c > 1.5, the radial motion of the TLV centre changes again away from the endwall. Second, for all J values, except near the TE, $U_{\theta ,leak}$ is positive, i.e., opposite to the direction of blade rotation (figure 19(g)). Furthermore, $U_{\theta ,leak}$ increases with s/c until the chordwise location of TLV rollup (figure 19(a–c)), and decreases thereafter. The sign of $U_{\theta ,leak}$ is counterintuitive since one would expect that the blade tip motion would induce a flow in the same direction within the narrow tip gap. Such a phenomenon could only be driven by pressure gradients, with the region of PS high pressure located close to the LE, and the SS minimum pressure located at higher s/c, consistent with the results of numerical simulations of this blade (Michael et al., Reference Michael, Choi and Otero2024).

Circumferentially downstream of the blade (figure 18, middle column), at least the two co-rotating secondary vortices, referred to earlier as SV1 and SV2, are visible in the shear layer for all the present J values. The mechanisms involved in their formation are similar to those described for J = 0.85 and are not repeated here. At s/c = 1.03, i.e., close to the TE, vortex SV1 travels slightly further upstream with decreasing J, while the location of SV2 essentially remains unchanged. Their entrainment into the TLV occurs earlier with increasing J (supplementary movies 4 and 7–9), most likely owing to the shorter distance to travel. For the case with the largest distance, J = 0.68, both secondary vortices start to break up while still located in the shear layer (supplementary movie 9). Their peak vorticity, which could be used for assessing their strength, appears to have the highest values at design J, and the lowest values occur at J = 0.68. The effect of their entrainment into the TLV can also be observed in the evolution of $\Gamma _{TLV}$ (figure 19(b)). While $\Gamma _{\textit{pos}}$ decreases at s/c > 1.00, $\Gamma _{TLV}$ continues to increase for the three higher J values, as the TLV entrains the secondary structures. A similar increase in TLV circulation circumferentially downstream of the blade of ducted propellers has been reported in Chesnakas and Jessup (Reference Chesnakas and Jessup2003) and Oweis et al. (Reference Oweis, Fry, Chesnakas, Jessup and Ceccio2006). At J = 0.68, $\Gamma _{TLV}$ decreases at 1.0 ≤ s/c ≤ 1.25, as the TLV begins to break up and spread, but increases again at s/c > 1.25, owing to the delayed entrainment of the shear layer and the secondary vortices. The impact of TLV breakup, especially at J = 0.68, can also be observed in the rapid reduction of peak $\langle \omega _{\theta }\rangle$ (figure 19(c)) starting from mid-chord, despite the overall increase in TLV circulation caused by the continued entrainment of the shear layer. It should also be noted that the earlier formation, breakup, as well as higher circulation of the TLV with decreasing flow rate have been seen in several other axial turbomachines tested in our facility (Li et al., Reference Li, Chen, Tan and Katz2019; Saraswat et al., Reference Saraswat, Koley, Joly and Katz2025). With decreasing J, the earlier and stronger TLV also enhances the formation and radially inward entrainment of the CRV originating from the endwall boundary layer (figure 18, left and middle columns). Before concluding this discussion, the right column of figure 18 demonstrates the effect of J on the distributions of $\langle \omega _{z}\rangle /{\Omega}$ circumferentially downstream of the blade. All the distributions contain the signature of the positive layer originating from the blade tip and the PS wake, and the negative layer originating from the blade SS which contains the quasi-axial vortices. The peak magnitudes of negative $\langle \omega _{z}\rangle$ are present in the vicinity of SV1 and SV2, but do not coincide with their $\langle \omega _{\theta }\rangle$ peaks. Hence, the previously discussed axial stretching of negative $\langle \omega _{z}\rangle$ is expected to occur in all cases.

Figure 20. First row: colour contours of 3 × 3 median-filtered $(\langle \omega _{z}\rangle \partial U_{z}/\partial z)/\varOmega ^{2}$ for J = 0.933: (a) r^* = 0.972, (b) r^* = 0.974, (c) r^* = 0.977. In panels (a–c), black lines are of $\langle \omega _{\theta }\rangle /\varOmega =40$ , and white markers indicate cavitation inception events. Second row: colour contours of $(\langle \omega _{z}\rangle \partial U_{z}/\partial z)/\varOmega ^{2}$ in regions where $\langle \omega _{z}\rangle$ < 0 and $\langle \omega _{z}\rangle \partial U_{z}/\partial z$ < 0: (d) s/c = 1.06, (e) s/c = 1.09, (f) s/c = 1.12. In panels (d–f), line contours: magenta- $\langle \omega _{\theta }\rangle /\varOmega =45$ , purple- $\langle \omega _{\theta }\rangle /\varOmega =30$ , yellow- $\langle \omega _{z}\rangle /\varOmega =-20$ . Grey horizontal line: r^* = 0.977. The black vertical lines bound the axial extent of cavitation inception events shown in panels (a–c). The superimposed in-plane velocity vectors are diluted 2:1 in both r and z for clarity.

Figure 20 shows a comparison between locations of cavitation inception and the ensemble-averaged flow structure for J = 0.933. In contrast to design J, most of the events are clustered in a single region located at 1.05 ≤ s/c ≤ 1.14 and 0.80 ≤ z/c _a ≤ 0.90 (figure 20(a–c)). This circumferential extent is in between the CI₁ and CI₂ regions of design J but is axially closer to the blade TE. Here again, the locations of inception appear to have a better agreement with regions of negative $\langle \omega _{z}\rangle$ and high negative $\langle \omega _{z}\rangle \partial U_{z}/\partial z$ (figure 20(a–c)). The perpendicular view (figure 20(d–f)) highlights the evolution of the stretched negative $\langle \omega _{z}\rangle$ in the relevant areas. At s/c = 1.06, as SV1 is getting entrained into the TLV, the regions containing peaks of both negative $\langle \omega _{z}\rangle$ , highlighted through a yellow line contour, and its stretching are located around both SV1 and SV2, and coincides with the cavitation inception domain highlighted through black vertical lines. At s/c = 1.09, the cavitation inception region partially overlaps with the region of peak axial stretching and elevated $\langle \omega _{z}\rangle$ , and the structures involved are the same as those for CI₂ at the design J. At s/c = 1.12, SV2 is located radially above the TLV, and the stretched axial vorticity, located to the right of both, still partially overlaps with the inception range. Note that the radial location of peak stretching also changes, implying that events appearing in the same cluster do not necessarily occur at the same r ^*.

The locations of cavitation inception for J = 0.803 and 0.68 are compared with those observed for higher J values in figure 21. At J = 0.803, while some events still appear as axially aligned bubbles at s/c > 1, they mostly occur within the TLV, and have a dominant scatter in the range 0.60 ≤ s/c ≤ 0.90 along its path. At J = 0.68, all the inception events occur along the TLV centre, most of them at 0.45 ≤ s/c ≤ 0.7, with the highest concentration centred around s/c ∼ 0.6. The earlier rollup and higher TLV strength (higher blade loading) with decreasing J appear to shift the location of cavitation inception further circumferentially upstream. The most likely location of inception along the TLV is close to the point of peak $\langle \omega _{\theta }\rangle$ (figure 19(c)), and does not occur at higher s/c, despite the increase in TLV circulation (figure 19(b)). This trend is presumably associated with the breakup of the TLV core and its fragmentation into multiple vortices along the aft part of the blade. Upon reduction of the mean pressure below the inception level, cavitation also expands to the secondary structures near the TE, in agreement with the persistence of the secondary vortices under all operating conditions (figure 18).

Figure 21. Distribution of the location of cavitation inception events for various advance ratios.

3.5. Discussion

The mean pressure around the blade tip TE is affected by the flow around the blade and the presence of the TLV. The RANS-predicted mean surface pressure distribution reported in Michael et al. (Reference Michael, Choi and Otero2024) shows that the SS blade pressure starts increasing from mid-chord to the aft of the blade. Furthermore, as the pressure increases with increasing distance from the TLV centre, the secondary vortices developing in the vicinity of the blade TE, but away from the TLV, are in a region of elevated mean pressure. This observation implies that the formation of localised regions with very low pressure outside of the TLV core can only occur owing to vortex stretching, and the resulting reduced radius and increased vorticity. We have already demonstrated that cavitation first occurs in regions where quasi-axial secondary vortices are stretched axially by the flow induced by the circumferentially aligned TLV and secondary vortices located in the shear layer separating the leakage flow from the main passage flow. This vortex stretching-induced cavitation inception, where a small, weak vortex is elongated by several larger and perpendicular vortices, is not unique to the present propeller flow. Therefore, it is instructive to compare the axial strain rates and vortex stretching magnitudes measured here with those reported in other flows exhibiting the same mechanism, such as turbulent shear flows (Katz & O’Hern Reference Katz and O’Hern1986; Agarwal et al., Reference Agarwal, Ram, Lu and Katz2023). For a turbulent shear layer behind a backward-facing step, Agarwal et al. (Reference Agarwal, Ram, Lu and Katz2023) show that at a flow speed of the same order of magnitude as the current U _T , cavitation first occurs in quasi-streamwise vortices that are stretched by the much larger and stronger spanwise vortices. Their most probable magnitude of axial strain rate is 500–1000 s⁻¹ but extends to more than 2500 s⁻¹ in extreme cases. The most probable vortex stretching term magnitude, along its axis, is of the order of 10⁵. For this propeller, statistics of axial vortex stretching for design J, in the cavitation inception region, i.e., r ^* > 0.95, and at s/c = 1.05, z/c _a = 0.7 to 0.8; and s/c = 1.13, z/c _a = 0.75 to 0.85, are provided in figure 22. Probability density functions (PDFs) of the instantaneous axial straining, $\partial u_{z}/\partial z,$ are presented in figure 22(a). The most probable value of ∼500 s⁻¹ and the peak value of ∼3000 s⁻¹ are similar in magnitude to those reported in Agarwal et al. (Reference Agarwal, Ram, Lu and Katz2023). Unfortunately, we have not measured the instantaneous $\omega _{z}$ , but the ensemble-averaged data obtained from the time-resolved measurements indicate that the magnitude of $\langle \omega _{z}\rangle$ is dominated by $\partial U_{\theta }/\partial r$ , i.e. $\partial U_{\theta }/\partial r\gg (\partial U_{r}/r\partial \theta +U_{\theta }/r)$ . Hence, we use $\partial u_{\theta }/\partial r$ as a surrogate for the instantaneous axial vorticity and presents its PDF in figure 22(b). As is evident, the magnitude of the negative values is larger, extending to −4000 s ^-1 . Furthermore, the negative surrogate vorticity at s/c = 1.13 has a higher magnitude than that at s/c = 1.05, as one would expect because of axial vortex stretching. The PDF of the surrogate axial vortex stretching term $(\partial u_{\theta }/\partial r)\partial u_{z}/\partial z$ , shown in figure 22(c), reaches peak negative values of 10⁶ which is an order of magnitude higher than the shear layer results of Agarwal et al. (Reference Agarwal, Ram, Lu and Katz2023). This difference explains the higher $\sigma _{i,{U_{T}}}$ (∼0.95) for the present propeller than those measured in the shear layer (∼0.5).

Figure 22. Probability density functions: (a) axial strain rate $\partial u_{z}/\partial z$ ; (b) surrogate axial vorticity $\partial u_{\theta }/\partial r$ ; and (c) surrogate axial vortex stretching, $\partial u_{z}/\partial z$ ( $\partial u_{\theta }/\partial r$ ), computed in the region of cavitation inception for J = 0.85. The sampling areas are r^* > 0.95 with 0.7 ≤ z/c_a ≤ 0.8 at s/c = 1.05, and r^* > 0.95 with 0.75 ≤ z/c_a ≤ 0.85 at s/c = 1.13. The upper horizontal axis shows dimensional values, and the lower horizontal axis shows the same data normalised using $\varOmega$ (for panels a and b) or $\varOmega$ ² (for panel c).

Figure 23. Effect of advance ratio on the distribution of TKE, $k/U_{T}^{2}$ , calculated from the phase-locked dataset at: (left) s/c = 0.75, (middle) s/c = 0.88, (right) s/c = 1.13. (a–c) J = 0.933, (d–f) J = 0.85, (g–i) J = 0.803 and (j–l) J = 0.68. Line contours, solid black: $\langle \omega _{\theta }\rangle /\varOmega =30$ , dashed black: $\langle \omega _{\theta }\rangle /\varOmega =-20$ .

The following discussion attempts to understand the shift in the location of cavitation inception below design conditions. As shown in the previous sub-section, changing the operating conditions alters the strength of the TLV, as well as the location of its rollup and breakup. Fragmentation of tip leakage vortices and the associated increase in turbulence have been seen in essentially all the ducted axial turbomachines tested in our facility (Wu et al., Reference Wu, Miorini, Tan and Katz2012; Li et al., Reference Li, Chen and Katz2017, Saraswat et al., Reference Saraswat, Koley, Joly and Katz2025). The mechanisms affecting the breakdown of vortices under the influence of adverse pressure gradients have been investigated in a variety of flows (e.g., Hall, Reference Hall1972; Escudier, Reference Escudier1988; Lucca-Negro and O’Doherty, Reference Lucca-Negro2001). Inherent to all axial pumps and compressors, there are adverse pressure gradients in the aft part of the rotor passage. As the TLV migrates to this region, it contracts along its axis, making it unstable and prone to breakdown. Once the so-called “spiral-type vortex breakdown” occurs (Lucca-Negro and O’Doherty, Reference Lucca-Negro2001), the TLV starts meandering, and its core fragments into multiple filaments that occupy a growing area. Consequently, the mean vortex size increases and its peak vorticity magnitude decreases. This effect becomes more pronounced below design conditions, where the blade’s peak loading and the point of TLV rollup shift towards the leading edge, and the adverse pressure gradients start earlier and become stronger. Consequently, the TLV breakdown occurs earlier, appears to increase in intensity and manifests as an increase in the vortex core’s turbulence level. Other mechanisms that affect the breakdown include migration of the TLV to the PS of the neighbouring blade, a phenomenon that is more prominent in multi-blade compressors and pumps, and flow non-uniformities caused by, e.g., other co-, counter-rotating, and even inclined secondary vortices (Lim, Reference Lim1998; Le et al., Reference Le, Borazjani, Kang and Sotiropoulos2011). A detailed discussion on the evolution of turbulence is beyond the scope of this paper. However, to demonstrate the effect of TLV breakup, figure 23 compares the distributions of TKE, calculated using the high-resolution, phase-locked dataset, at s/c = 0.75, 0.88 and s/c = 1.13. Circumferentially upstream of the TE (s/c = 0.75 and 0.88), there is strikingly high turbulence at J = 0.68 in comparison with the other advance ratios, which is associated with earlier breakdown of vortices. With decreasing J, the regions of elevated turbulence expand from only the shear layer at J = 0.933 to the CRV and the vicinity of the TLV at J = 0.85, then to the centre of the TLV at J = 0.803, and finally to a broad area with much higher values at J = 0.68. For J = 0.68, the TKE is particularly high in the shear layer, the TLV centre and the entrained separated backflow boundary layer. At s/c = 1.13, the areas with elevated turbulence expand in all cases, with additional peaks appearing in the vicinity of SV1, SV2 and the blade wake. The regions of cavitation inception at J = 0.933 and 0.85, which are between the TLV and SV1, and between SV1 and SV2, are not characterised by particularly high turbulence. This observation indicates that the onset of cavitation preferentially occurs in regions where the flow structure is at least ‘quasi-coherent’. For the cases where the vicinity of SV1 and SV2 becomes turbulent, i.e., the lower two J values, the cavitation inception events shift to the TLV. These observations are also consistent with measurements of cavitation inception in turbulent shear layers (Agarwal et al., Reference Agarwal, Ram, Lu and Katz2023), where the likely inception site is in a domain where secondary vortices are quasi-coherent, upstream of the regions with maximum turbulence level. A similar trend is observed along the TLV core, where inception occurs closer to the TE at J = 0.803, but at J = 0.68 shifts further upstream along the TLV, circumferentially before the TLV breakdown, as indicated by a reduction in peak $\langle \omega _{\theta }\rangle$ (figure 19(c)) and elevated turbulence near its centre.