3.1. Qualitative description of the flow

In this section, the global features of the flow are described. Figures 4( $a$ ) and 4( $b$ ) show photographs of developed vortex cavitation from the experiments of Chesnakas & Jessup (Reference Chesnakas and Jessup2003). Annotations on the figure detail the key flow features identified by Chesnakas & Jessup (Reference Chesnakas and Jessup2003) and Oweis et al. (Reference Oweis, Fry, Chesnakas, Jessup and Ceccio2006a ). The photograph is oriented looking down at the blade tip such that the blade rotation causes the tip to move upwards in the frame. The bulk flow through the duct moves in the positive $x$ -direction, from left to right. The pressure gradient across the blade results in a leakage flow moving in the negative $x$ -direction in the gap region between the blade and the duct. The leakage flow separates off the suction side blade tip and rolls up into a coherent vortex, termed the primary leakage vortex hereafter. The primary leakage vortex core is seen to coincide with regions of vapour upstream from the trailing edge. Aft of the trailing edge, an additional vapour region coincides with the merger of secondary vortex structures from the trailing edge wake. The extent of the trailing edge vortex cavity differs between figures 4( $a$ ) and 4( $b$ ) suggesting that the trailing edge vortex cavitation is an intermittent process.

Figure 4( $c$ ) shows an isosurface demarcating a low-pressure coefficient region of $C_p \lt -3.9$ from the current LES simulation results. The pressure coefficient is defined as

(3.1) \begin{eqnarray} C_p = \frac {p-p_{\infty }}{\frac {1}{2}\rho U_{\infty }^{2}}, \end{eqnarray}

where $p$ is the local pressure, $p_{\infty }$ is the reference pressure, $U_{\infty }$ is the free stream velocity and $\rho$ is the fluid density. The reference pressure is taken as the time-averaged pressure inside the duct at midspan upstream of the rotor at $x=-3.5R$ . The $C_p$ isosurface in figure 4( $c$ ) is coloured by vorticity magnitude. The entire low-pressure region corresponds to a vorticity magnitude $\lvert \omega \rvert R/U_\infty \gt 200$ . The low-pressure region follows a qualitatively similar trajectory to the primary leakage vortex observed to cavitate in figures 4( $a$ ) and 4( $b$ ). Additional qualitative similarities are seen in the orientation and location of secondary vortical structures of higher vorticity magnitude $\lvert \omega \rvert R/U_\infty \gt 600$ merging in the blade wake aft of the trailing edge. An in-depth discussion of the small-scale vortex structures near the trailing edge is presented later in § 3.3.

Figure 4. ( $a,b$ ) Photograph of developed tip vortex cavitation from the experiments of Chesnakas & Jessup (Reference Chesnakas and Jessup2003) at $\sigma =2(p_{\infty } - p_v)/(\rho U_{\infty }^{2})=5.6$ . ( $c$ ) Isosurface of instantaneous $C_p=-3.9$ from the current LES results on Grid 2 coloured by vorticity magnitude $\lvert \omega \rvert R/U_\infty$ .

3.2. Validation

In this section, the time-averaged loads and flow fields from the LES simulations are validated against the experiments of Chesnakas & Jessup (Reference Chesnakas and Jessup2003), Oweis (Reference Oweis2003) and Oweis et al. (Reference Oweis, Fry, Chesnakas, Jessup and Ceccio2006a , Reference Oweis, Fry, Chesnakas, Jessup and Cecciob ). Three-dimensional LDV data are provided by NSWCCD (Thad Michael, personal communication).

3.2.1. Loads

The time-averaged load statistics from the LES at an advance ratio of $J=0.98$ are compared with the experiments of Chesnakas & Jessup (Reference Chesnakas and Jessup2003), Oweis (Reference Oweis2003), Oweis et al. (Reference Oweis, Fry, Chesnakas, Jessup and Ceccio2006a ) and Oweis et al. (Reference Oweis, Fry, Chesnakas, Jessup and Ceccio2006b ). The advance ratio is defined as

(3.2) \begin{eqnarray} J = \frac {2\pi U_{\infty }}{\Omega D}, \end{eqnarray}

where $U_{\infty }$ is the free stream velocity, $\Omega$ is the rotational velocity in units of radians per second and $D$ is the propeller diameter. The experiments of Chesnakas & Jessup (Reference Chesnakas and Jessup2003), Oweis et al. (Reference Oweis, Fry, Chesnakas, Jessup and Ceccio2006a ) and Oweis et al. (Reference Oweis, Fry, Chesnakas, Jessup and Ceccio2006b ) adopted the time-average velocity value midway between the hub and duct in the radial direction as $U_{\infty }$ . Note that $U_{inf,BC}$ prescribed at the simulation inflow is chosen such that the volumetric flow rate through the duct reproduced that of the corresponding experiments, as discussed in the preceding section. Table 2 summarises the mean loads, comparing the LES simulations to various experiments for propeller P5407. The thrust and torque coefficients are defined as

(3.3) \begin{eqnarray} K_T=\frac {4\pi ^2 T}{\rho \Omega ^2 D^4} ,\quad K_Q= \frac {4\pi ^2 Q}{\rho \Omega ^2 D^5}, \end{eqnarray}

where $T$ is the thrust force, $Q$ is the torque and $\rho$ is the fluid density. The difference between the LES loads on Grid 1 and Grid 2 is $1\,\%$ for $K_T$ and less than that for $K_Q$ , suggesting grid convergence of the integral circulation. The experimental load data for $K_T$ from three different experiments contains some scatter with a difference of approximately $13\,\%$ of the $K_T$ magnitude. For $K_Q$ , one of the experimental values has a minor $2\,\%$ difference. For both Grids 1 and 2, the LES shows agreement with the experimental $K_Q$ values and lying with in the scatter for $K_T$ .

Table 2. Comparison of load statistics for Propeller P5407 at $J=0.98$ and $Re_{tip}=1.1\times 10^{6}$ . Here EXP-1 is Chesnakas & Jessup (Reference Chesnakas and Jessup2003), EXP-2 is Oweis (Reference Oweis2003) and EXP-3 is Oweis et al. (Reference Oweis, Fry, Chesnakas, Jessup and Ceccio2006b ). Here LES-G1 is the current simulation on Grid 1 and LES-G2 is the current simulation on Grid 2. Data from Chesnakas & Jessup (Reference Chesnakas and Jessup2003) was provided by NSWCCD (Thad Michael, personal communication).

Loads are also measured at an advance ratio around $J=1.1$ for P5407 and compared with the experimental data of Chesnakas & Jessup (provided by Thad Michael, personal communication). Measurements of the inflow were not available, making it difficult to precisely determine the duct mass flow and $U_{\infty }$ for the calculation of the advance ratio, $J$ . For that reason, a range of inflows velocities, $U_{inf,BC}$ are considered for the same propeller rotation rate corresponding to roughly $J=1.1$ . The data are plotted in figure 5 and shows that a 1.5 % reduction in $U_{inf,BC}$ results in a 13 % reduction in the thrust coefficient $K_T$ . Such sensitivity ${\rm d}K_T/{\rm d}J$ matches the experimental results well. The sensitivity is also well predicted for ${\rm d}K_Q/{\rm d}J$ .

These results highlight the extreme load sensitivity to the inflow of ducted propulsors and to make the case that integral quantities such as volumetric flow rate are preferred relative to single-point inflow measurements for definition of the advance ratio $J$ . This conclusion is especially important when comparisons are being made across different test set-ups with significantly different Reynolds numbers. Reynolds number variations modify the boundary layer thicknesses on both the shaft and the duct wall and may therefore result in a different midchannel velocity for the same mass flow rate. The same effect may also occur when changing propeller rotation rates within the same test set-up. Lower rotation rates than the ‘nominal’ condition at which the inflow is measured, such as that considered here, result in a reduction in the favourable pressure gradient experienced by the upstream boundary layer on both the duct and shaft. Weaker favourable pressure gradients result in a thickening of the boundary layer, causing the peak velocity at midspan to increase assuming the mass flow rate remains constant.

Figure 5. Force coefficients $K_T$ and $K_Q$ across range of $J$ . The LES results for Grid 1 are superimposed on data from Chesnakas & Jessup (personal communication).

3.2.2. Grid refinement and LDV comparisons

The time-averaged flow statistics from the LES are compared with the three-dimensional LDV experimental data of Chesnakas & Jessup (Reference Chesnakas and Jessup2003) with extracts taken in the $x{-}r$ plane where the axial coordinate $x$ and radial coordinate $r$ are normalised by the propeller radius $R$ . Chesnakas & Jessup (Reference Chesnakas and Jessup2003) state that the uncertainty in velocity measurements was $10\,\%$ of $U_{\infty }$ at the vortex core location and in the blade wake. The cause of uncertainty was linked to inadequate sample sizes taken in regions of high turbulence. Figure 6 shows the LDV grid in figure 6( $a$ ) compared with LES numerical Grid 1 (figure 6 $b$ ) and Grid 2 (figure 6 $c$ ). The LES solution mesh extends farther towards the duct wall than the LDV with points clustered in the duct boundary layer.

Figure 6. Grid resolutions in the $x{-}r$ plane at the blade trailing edge for ( $a$ ) the LDV measurements of Chesnakas & Jessup (Reference Chesnakas and Jessup2003), ( $b$ ) the present coarse LES Grid 1 and ( $c$ ) present LES Grid 3.

Figure 7 compares contour slices of the mean axial velocity $\langle u_x \rangle /U_\infty$ in the $x{-}r$ plane at the blade trailing edge ( $R\theta /c=0$ ). The axial flow moves upstream (negative $u_x$ in blue) above the propeller blade tip radius, which is termed the tip leakage flow. The axial flow moves in the positive $x$ direction at values of the radial coordinate below the blade tip ( $r/R\lt 1$ ). There are discrepancies in the LES and LDV results near the duct surface where the experimental viewing window protrudes into the circular duct cross-section and the LDV has a coarser resolution. The LES predicts a thinner boundary layer on the duct wall and larger magnitude of the reversed leakage flow velocity near $r/R=1.01$ . Some blade-to-blade scatter is seen in the extracted profiles of the LDV data. Grid convergence is seen between the three mesh resolutions in the mean axial flow velocity.

Figure 7. Contours of the mean axial velocity $\langle u_x \rangle /U_\infty$ in the $x{-}r$ plane at the blade trailing edge for ( $a$ ) the LDV measurements of Chesnakas & Jessup (Reference Chesnakas and Jessup2003), ( $b$ ) the present LES on Grid 1 and ( $c$ ) present LES on Grid 3 profiles are extracted at $x/R=$ 0.145, 0.150, 0.154, 0.158 and 0.162 are shown in ( $d$ ) for LES Grid 1 (), LES Grid 2 (), LES Grid 3 (), LDV Blade 1 (), LDV Blade 2 (), LDV Blade 3 ().

Figure 8 compares the azimuthal velocity $u_\theta /U_\infty$ in the inertial frame. The $x{-}r$ plane at the blade trailing edge is shown in $a{-}c$ . The LDV measurements and LES simulations with both grid resolutions show $u_\theta /U_\infty \approx 1$ in the leakage vortex lower periphery and separation sheet. The LDV measurements predict slightly negative azimuthal velocity in the vortex upper periphery and leakage flow, while both LES solutions show values near zero. The LES results underpredict the azimuthal velocity beneath the leakage vortex ( $r/R \lt 0.94$ ) relative to the experimental measurements. In figure 8( $d$ ), profiles are extracted in the axial direction at $R\theta /c = 0.05$ to avoid the blanking region where the blade trailing edge obscures optical access. Good agreement is seen in the blade wake velocity deficit, though the blade wake thickness is overpredicted by LES relative to the LDV measurements.

Figure 8. Contours of the mean azimuthal velocity $\langle u_\theta \rangle /U_\infty$ in the $x{-}r$ plane at the blade trailing edge for ( $a$ ) the LDV measurements of Chesnakas & Jessup (Reference Chesnakas and Jessup2003) ( $b$ ) the present LES on Grid 1 and ( $c$ ) present LES on Grid 3. Profiles extracted at $R\theta /c = 0.05$ and $r/R=$ 0.944, 0.95, 0.968, 0.989 and 0.995 are shown in ( $d$ ) for LES Grid 1 (), LES Grid 2 (), LES Grid 3 (), LDV Blade 1 (), LDV Blade 2 (), LDV Blade 3 () .

The evolution of trailing vortices behind the blade trailing edge is shown in figure 9. Figure 9(a,d,g,j) show the measurements of mean azimuthal vorticity $\langle \omega _\theta \rangle R/U_\infty$ behind Blade 2 from the experiments of Chesnakas & Jessup (Reference Chesnakas and Jessup2003) at $R\theta /c =$ 0.0, 0.084, 0.167 and 0.251. Figure 9(b,e,h,k) show the time average results on Grid 1 and figure 9(a,d,g,j) show the time average results from Grid 3 in the present LES. The vorticity is mostly directed in the positive $\theta$ direction (out of the page or anticlockwise using the right-hand rule) and peaks at the leakage vortex centre. The leakage vortex is formed from the rollup of the leakage flow separating off the blade suction side at a specific chord location before the blade trailing edge. The leakage vortex is therefore offset from the blade trailing edge into the negative axial direction. A separation sheet connects the leakage vortex core to the blade tip, visible as a narrow region of high vorticity of the same sign as the leakage vortex core. Wall shear of the reversed leakage flow induces counter-rotating vorticity on the duct wall (seen in orange). To the left of the leakage vortex core, the region of counter-rotating vorticity begins to grow away from the duct wall. The LDV data does not have sufficient resolution to capture the near-wall shear and this region of counter-rotating vorticity. Farther behind the blade at $R\theta /c =$ 0.084, a second vortex corotating vortex is seen to the right of the primary leakage vortex in both the LDV (figure 9 $d$ ) and LES (figure 9 $e,f$ ). This secondary vortex forms behind the blade trailing edge in the rollup of the suction side blade wake and separation sheet. The trailing edge vortex is weaker than the primary leakage vortex. At $R\theta /c =$ 0.167, the trailing edge vortex is directly on top of the leakage vortex core in both the LDV (figure 9 $g$ ) and LES (figure 9 $h,i$ ). The leakage vortex is flattened, with a larger region of high vorticity on the upstream (left-hand) side of the core than the right-hand side. Finally, at $R\theta /c =$ 0.251 the trailing edge vortex begins to merge with the leakage vortex from the upstream periphery in both LDV (figure 9 $j$ ) and LES (figure 9 $k,l$ ) results. The counter-rotating vortex has separated from the duct wall in the LES. Overall, very strong agreement in the evolution of the trailing vortex system is seen between the LDV measurements and LES. Both LES grid resolutions predict similar rotation strengths and core trajectories for both the leakage and trailing edge vortex. The behaviour of the counter-rotating vorticity on the duct wall is also reproduced closely between the two grid resolutions shown.

Figure 9. Contours of the mean azimuthal vorticity field in the $x{-}r$ planes behind the blade trailing edge at $R\theta /c = 0.00$ for ( $a,b,c$ ), 0.084 ( $d,e,f$ ), 0.167 ( $g,h,i$ ), 0.251 ( $j,k,l$ ). Panels (a,d,g,j) show the LDV measurements from Chesnakas & Jessup (Reference Chesnakas and Jessup2003) while (b,e,h,k) show the present LES on Grid 1 and (c,f,i,l) show present LES Grid 3.

For a more detailed comparison with the experiment, profiles are taken of $\langle \omega _{\theta } \rangle R/U_\infty$ in the $x{-}R$ plane at the blade trailing edge (figure 10 $a{-}c$ ). The profiles move along the radial direction at various axial locations near the primary leakage vortex core. The three LES mesh resolutions are plotted as solid lines while the three individual blades from the LDV are plotted as symbols. The mean $\omega _{\theta }$ peaks correspond closely and the LES results are within the experimental scatter. Differences near the duct wall surface are observed as previously discussed.

Figure 10. Profiles of the mean azimuthal vorticity $\langle \omega _\theta \rangle R/U_\infty$ in the $x{-}r$ plane at the blade trailing edge. Profiles extracted at $x/R=$ 0.145, 0.150, 0.154, 0.158 and 0.162 are shown in ( $d$ ) for LES Grid 1(), LES Grid 2(), LES Grid 3 (), LDV Blade 1 (), LDV Blade 2 (), LDV Blade 3 ().

A comparison of the locations of the tip leakage vortex centre is shown in table 3. The leakage vortex centre is taken to be the location of the maximum mean vorticity. The vortex locations for the three individual blades in the LES mesh are all within $1\,\%$ of each other, evidence of statistical convergence. Some scatter is observed in the experimental results between the blades, with differences up to $6\,\%$ in the axial direction. The LES results agree best with the experimental results for Blade 3 within 1.3 % for the axial location and 0.6 % for the radial location.

As an assessment of the appropriateness of the mesh resolution, figure 11 shows a contour of eddy-viscosity $\nu _{T}$ normalised by the molecular viscosity $\nu$ in the tip vortex region at the trailing edge ( $R\theta /c=0$ ). As seen in (2.1) and (2.2), the eddy viscosity is the parameter by which the subgrid-scale model influences the LES solution. The eddy viscosity ratio is therefore a measure of the unresolved turbulent scales and tends to zero as the grid is refined. Figure 11 shows that in the tip vortex interaction region, $\nu _{T}/\nu$ has a maximum of approximately $6$ for Grid 1, $3$ for Grid 2 and $2$ for Grid 3. This suggests sufficient resolution in the vortex interaction areas to minimise the LES subgrid model’s influence on the observed flow structure.

Table 3. The $x{-}r$ plane mean tip leakage vortex centre locations and strength of the experiment of Chesnakas & Jessup (Reference Chesnakas and Jessup2003) Here EXP-1 compared with LES results for Grid 1 (LES-G1) and Grid 2 (LES-G2) grids.

Figure 11. A contour slice in the $x{-}r$ plane at the propeller trailing-edge tip $R\theta /c=0$ showing the LES eddy-viscosity $\nu _{T}$ normalised by the molecular viscosity $\nu$ for ( $a$ ) Grid 1, ( $b$ ) Grid 2 and ( $c$ ) Grid 3.

3.3. Instantaneous flow field

This section provides an in-depth analysis of tip vortex flow structures and their relation to low-pressure events that drive cavitation inception. The instantaneous LES results on the finest discretisation (Grid 3) are examined.

Assessment of the instantaneous pressure field reveals the primary leakage vortex has an identifiable low-pressure region, visualised with an isocontour of $C_p=-3.7$ in figure 12. No coherent trailing edge vortex is captured in the trailing edge region by visualisation of the pressure field. However, side lobes of low pressure are seen wrapping helically around the primary leakage vortex core both forward and aft of the blade trailing edge. In figure 12( $a$ ), these side lobes are seen to correspond to areas of strong axial vorticity, while the primary leakage vortex core mostly aligns with the azimuthal direction, $\theta$ . The high axial vorticity is evidence that the side lobes seen in the pressure isosurface correspond to secondary vortex filaments with a core inclined with respect to the primary leakage vortex. This observation from the LES is consistent with high-speed video of cavitation inception from experimental tests of a similar propulsor by Saraswat et al. (Reference Saraswat, Panigrahi and Katz2024). In figure 12( $b$ ), an isosurface of $C_p=-5.5$ is displayed in darker blue along with the transparent isocontour of $C_p=-3.7$ in cyan. It reveals that the leakage vortex core pressure continues to decrease aft the blade trailing edge in the $\theta$ direction. At the bottom of the image ( $R\theta /c=0.4$ ), a helical oscillation mode of the primary leakage vortex core is observed behind the blade.

Figure 12. ( $a$ ) Instantaneous isosurface of $Cp=-3.7$ coloured by axial vorticity $\omega _{x}R/U_\infty$ . ( $b$ ) Instantaneous isosurfaces of $C_p = -5.5$ and $-3.7$ are displayed transparently and coloured by $C_p$ .

Figure 13 displays an isosurface of $C_p = -4.7$ (transparent grey) along with isocontours of $Q$ -criterion, revealing a more complex flow field of small-scale vortex structures. The $Q$ -criterion is the second invariant of the velocity gradient tensor (Hunt et al. Reference Hunt, Wray and Moin1988), defined as

(3.4) \begin{eqnarray} Q = \frac {1}{2} \lvert \boldsymbol {\omega } \rvert - S_{ij}S_{ij} = \frac {\partial u_i}{\partial x_j}\frac {\partial u_j}{\partial x_i}, \end{eqnarray}

where $S_{ij}$ is the velocity strain rate tensor and $\boldsymbol {\omega }$ is the vorticity vector. For incompressible flows, it can also be shown that

(3.5) \begin{eqnarray} Q = \frac {1}{2\rho }\nabla ^2 p, \end{eqnarray}

demonstrating that $Q$ behaves as a source term for pressure (Dubief & Delcayre Reference Dubief and Delcayre2000). Figure 13 shows that the strongest instantaneous vortex structures are not colocated or aligned with the primary leakage vortex. The non-locality of high positive $Q$ with global pressure minima is a consequence of the elliptic nature of the pressure Poisson equation, which results in changes to the pressure field acting over larger distances than the kinematic source term. This observation highlights the importance of rotation across both small and large scales to the lowest pressures driving cavitation inception.

In figure 13( $a$ ), isosurfaces of $Q R^2/U_{\infty }^2= 180\,000$ are coloured by the local pressure coefficient $C_p$ . The trailing edge wake remains at a pressure higher than ambient ( $p_\infty$ ) and much higher than the pressure in the primary vortex core. This is evidence that, instantaneously, there are no coherent vortex structures at the trailing edge with strong enough rotation to induce a distinct low-pressure core. In figure 13( $b$ ), isosurfaces of $Q$ are coloured by the sign of the azimuthal vorticity $\omega _\theta R/U_\infty$ , revealing that the flow separating off the trailing edge is a mixture of small-scale vortices corotating (red) and counter-rotating (blue) with the primary leakage vortex. These observations of the instantaneous LES results counter the description of the mean flow field given by Chesnakas & Jessup (Reference Chesnakas and Jessup2003) and shed additional insights on the descriptions of the instantaneous flow field given by Oweis et al. (Reference Oweis, Fry, Chesnakas, Jessup and Ceccio2006a ). In figure 13( $c$ ), isosurfaces of $Q$ are coloured by the sign of the axial vorticity $\omega _x R/U_\infty$ , and reinforce the observation of mixed positive and negative directions of rotation at the smallest scales.

Figure 13. Instantaneous isosurfaces of $ Q R^2/U_{\infty }^2=180000$ coloured in ( $a$ ) by the pressure coefficient $C_p$ , ( $b$ ) by the azimuthal vorticity $\omega _{\theta }R/U_\infty$ and ( $c$ ) by axial vorticity $\omega _{x}R/U_\infty$ . An isosurface of $C_p = -4.7$ is also displayed in transparent grey to denote the location of the primary leakage vortex.

The leakage flow aft of the midchord has a positive $u_{\theta }$ and negative $u_{x}$ and thus moves from northeast to southwest as denoted by the black arrow in figure 13(b). Vortices generated in the blade tip boundary layer are seen in figures 13( $b$ ) and 13( $c$ ) to be oriented perpendicular to the leakage flow. Small-scale vortex filaments are present throughout the separation sheet connecting the blade tip to the primary leakage vortex core aft of the core separation point. Interestingly, the separation sheet is seen to have instantaneous vortex structures that are aligned nearly parallel to the leakage flow. This observation indicates that the separation sheet is not simply transporting vorticity from the blade tip boundary layer. Rather, the shear of the separation sheet itself (discussed further in § 3.5.1) is dynamically relevant in the sense that the tip boundary layer vorticity gets tilted and deformed before being fed to the primary leakage vortex core. The orientation and placement of those leakage flow-aligned vortices in the separation sheet correspond to the previously discussed side lobes of low-pressure wrapping helically around the primary vortex core (figure 12).

After the blade passes, the vortex structures of the separation sheet and trailing edge wake then traverse over the top of the primary leakage vortex before meeting the incoming bulk flow through the duct, which moves from left to right in figure 13. The traverse process is influenced by the induced velocity of both the primary leakage vortex and its image, given the close proximity of the duct wall. Secondary vortex structures therefore have a short residence time in the region directly above the leakage vortex radially.

There is also an additional counter-rotating vortex visible to the left-hand side of the primary leakage vortex in figure 13( $b$ ). This counter-rotating vortex is located upstream in the axial direction and above the primary leakage vortex core in the radial direction and oriented nearly parallel to the primary vortex core. This upstream vortex results from the rollup of the duct boundary layer vorticity that occurs as the reversed leakage flow ( $u_x \lt 0$ ) meets the turbulent inflow ( $u_x \gt 0$ ). Similar behaviour was also observed in the waterjet pump experimentally studied by Wu et al. (Reference Wu, Miorini, Tan and Katz2012). To examine this rollup further, the azimuthal vorticity field is plotted on $x{-}r$ slices in figure 14( $a$ ). The primary leakage vortex is seen as a large-scale region of positive signed vorticity on all slices. On slices forward of the blade trailing edge, the duct boundary layer is visible as a very thin sheet of negative signed vorticity at the domain’s outer radial extent. Thus, the counter-rotating vorticity originates in the duct boundary layer in the reversed leakage flow. As the blade rotates (into the page), the adverse pressure gradient driving the leakage flow is removed. The duct boundary layer vorticity rolls up into a coherent vortex and migrates down from the duct wall farther behind the blade trailing edge. As the counter-rotating vorticity moves off the wall, it breaks up into smaller scale structures, which become intertwined with the small-scale positive signed vorticity of the separation sheet. The time taken for counter-rotating vortices to move downwards in the radial direction and for separation sheet vortex structures to traverse over top of the leakage vortex results in the small-scale interactions of corotating and counter-rotating vortices occurring between $20\,\%$ and $40\,\%$ of the chord length aft of the blade trailing edge. At the farthest slice aft of the blade trailing edge, most of the strong corotating vorticity has merged into the primary leakage vortex core. The vortex merger process observed in the present LES closely resembles that of the axial waterjet pump studied by Wu et al. (Reference Wu, Miorini and Katz2011a , Reference Wu, Tan, Miorini and Katzb , Reference Wu, Miorini, Tan and Katz2012). The breakup of negative signed azimuthal vorticity into smaller scales as a result of its interaction with the positive signed azimuthal vorticity of the separation sheet also mirrors the conclusions of Ortega et al. (Reference Ortega, Bristol and Savaş2003).

The interactions of vortex structures are quantified using the decomposition of the vorticity transport equation in stretching, tilting and diffusion terms of Zhang et al. (Reference Zhang, Shen and Yue1999). The vortex stretching term is $\boldsymbol {\omega \cdot S \cdot \hat {e}_\omega }$ where $\boldsymbol {\omega }$ is the vorticity vector, $\boldsymbol {S}$ is the strain rate tensor and $\boldsymbol {\hat {e}_\omega }$ is the unit vector aligned with the local axis of vorticity. Positive values indicate vorticity undergoing axial extension, while negative values are indicative of vortex compression. Vortex stretching is plotted in figure 14( $b$ ). Large positive stretching magnitudes are seen throughout the blade boundary layer and separation sheet forward of the trailing edge ( $R\theta /c \leqslant 0$ ), as is intuitively expected given the large velocity gradients in that region. Immediately aft of the blade trailing edge ( $R\theta /c = 0.11)$ , the thin region of sustained high stretching of the separation sheet scatters into smaller scale regions. At the second slice aft the trailing edge ( $R\theta /c = 0.21$ ), the highest vortex stretching occurs near the duct wall in the region coinciding with the separation sheet vortex structures traversing over the primary leakage vortex and meeting the counter-rotating vorticity of the leakage flow duct boundary layer. Farther aft at $R\theta /c = 0.32$ , where negative-signed vorticity from the leakage boundary layer has migrated off the duct wall, strong vortex stretching is seen again at the upstream periphery of the primary leakage vortex. This strongest vortex stretching far behind the blade occurs in an area nearer to the persistent low pressure of the primary vortex core than the strong stretching observed farther forward in the separation sheet. This observation highlights that extreme low-pressure events leading to cavitation inception are a multiscale phenomenon driven by both highly local stretching between corotating and counter-rotating small-scale vortices and the more persistent low-pressure regions driven by rotational flow at larger scales.

Figure 14. Instantaneous flow field on $x{-}r$ slices at $R\theta /c=-0.21,-0.11,0.00,0.11,0.21,0.32$ and 0.43. Colour contours are ( $a$ ) azimuthal vorticity component $\omega _{\theta }R/U_\infty$ , ( $b$ ) vortex stretching and ( $c$ ) vortex tilting.

To the authors’ knowledge, this is the first time that the cavitation inception mechanisms hypothesised by Chesnakas & Jessup (Reference Chesnakas and Jessup2003) have been evaluated quantitatively in the literature. More specifically, the present results find support for their hypothesis in the following ways: (i) strong vortex stretching persists far downstream ( $R\theta /c \gt 0.30$ ) to where inception was observed; (ii) strongest stretching driving inception occurs at small length scales. In addition, the present results extend the previous hypotheses of ducted propulsor cavitation by highlighting that the leakage flow duct boundary layer rollup is an important source of secondary vorticity, while Chesnakas & Jessup (Reference Chesnakas and Jessup2003) highlights the role of trailing edge vortex shedding.

Figure 14( $c$ ) shows the vortex tilting term $\boldsymbol {\omega \cdot S \cdot \hat {e}_t}$ . The vortex tilting is highest in the separation sheet aft the primary leakage vortex separation point, providing quantitative evidence to support the qualitative claims about blade tip vorticity realignment made in the discussion of figure 13. Unlike the stretching term, the vortex tilting term does not recover similar magnitudes to the separation sheet in the vortex interactions that occur farther behind the blade trailing edge.

The overall picture of vortex interactions in the wake of a ducted propulsor suggested by LES consists of corotating and counter-rotating small-scale vortices from the leakage flow, separation sheet, trailing edge wake and induced upstream vortex. A schematic illustration of the significant vortex flow structures suggested by the current LES is presented in figure 15( $a$ ). The blade is viewed from above the tip looking down at the trailing edge such that the blade rotates upwards in the plane of the page. The primary leakage vortex is illustrated in green and detaches from the blade upstream from the trailing edge ( $A$ ). Aft of the primary leakage vortex separation, additional vorticity of the blade tip boundary layer is shed and forms a separation sheet ( $B$ ). The separation sheet is composed of vortices both perpendicular to the leakage flow (red) and parallel to the leakage flow (blue). At the trailing edge, additional vorticity is shed from the blade pressure side boundary layer ( $C$ ). After the blade passes, the vortex structures of the separation sheet and boundary layer move above the primary leakage vortex ( $D$ ). The rollup of the leakage flow duct boundary layer induces a counter-rotating vortex parallel to the primary vortex and upstream in the axial direction, which is shown in yellow ( $E$ ). Far behind the blade trailing edge, vortex structures from the separation sheet and duct boundary layer rollup break up into smaller scales before eventually merging with the primary leakage vortex at the upstream periphery ( $F$ ). Figure 15( $b$ ) repeats figure 13( $c$ ), but showing the Grid 2 solution rather than Grid 3. The coarser grid solution is presented here to remove the chaotic smaller scale turbulent structures for clarity. The same viewing angle as the schematic illustration in figure 15( $a$ ) is used in figure 15( $b$ ). The separation sheet vortex structures both aligned and perpendicular are clearly visible. The complex trailing edge wake shows vortical structures of opposing signs intertwined.

Figure 15. ( $a$ ) Schematic of significant vortex structures in the ducted propeller flow near peak efficiency. Green represents the primary leakage vortex; red represents vortices perpendicular to the leakage flow originating in the blade boundary layer; blue represents the leakage-flow aligned vortices of the separation sheet; yellow is the induced counter-rotating vortex. ( $b$ ) Isosurfaces of instantaneous $Q R^2/U_\infty ^2 = 170000$ from the LES Grid 2 solution coloured by the axial vorticity. An isosurface of $C_p=-4.7$ is also shown to denote the primary leakage vortex core.

This section has provided a more detailed picture of the instantaneous tip vortex flow field than that provided by Chesnakas & Jessup (Reference Chesnakas and Jessup2003) and Oweis et al. (Reference Oweis, Fry, Chesnakas, Jessup and Ceccio2006a ). A large-scale coherent vortex emanating from the blade trailing edge is not seen in the instantaneous LES pressure field or $Q$ -criterion. Rather, instantaneous vortex structures aligned with the tip leakage flow were observed throughout the separation sheet connecting the separated primary leakage vortex core to the blade tip boundary layer. These leakage flow-aligned structures are observed both forward and aft of the blade trailing edge and are found to correlate with side lobes of low-pressure wrapping around the primary vortex core. This suggests that the trailing edge vortex discussed by Chesnakas & Jessup (Reference Chesnakas and Jessup2003) and Oweis et al. (Reference Oweis, Fry, Chesnakas, Jessup and Ceccio2006a ) is not generated by the blade trailing edge itself but within the separation sheet. Despite offering a somewhat different description of the relevant mechanisms driving the flow, the current LES results find agreement with the experimental observation that secondary vortex structures near the blade trailing edge were likely to cavitate during the merger process with the primary vortex, as evidenced by the statistics of low-pressure events, that are presented in § 3.4.

Figure 16. The joint PDF of low-pressure events in $x-\theta$ space for the LES solution on Grid 3 for ( $a$ ) $C_p \lt -6.2$ , ( $b$ ) $C_p \lt -7.2$ and ( $c$ ) $C_p \lt -7.8$ . The blade outline is shown as a solid black line.

3.4. Low-pressure event statistics

To estimate the probable location of cavitation inception, locations of extreme low-pressure events are collected for Grid 3 over 3.5 propeller revolutions. Low-pressure events are defined as any simulation control volume with an instantaneous $C_p$ less than some threshold value and is sampled every 40 simulation time steps to ensure events are statistically independent in time. The joint probability density function (PDF) in $x-\theta$ space is shown in figure 16 for various threshold values: (figure 16 $a$ ) $C_p \lt -6.2$ , (figure 16 $b$ ) $C_p \lt -7.2$ and (figure 16 $c$ ) $C_p \lt -7.8$ . The joint PDF is defined discretely by binning low-pressure events into discrete regions of $x-\theta$ space and dividing the number of samples in each bin by the total number of samples and the bin area. The normalisation of the PDF is completed separately for each threshold value of $C_p$ . Lower pressure events are increasingly rare and, as a result, show less smooth distributions in the joint PDF for a given simulation sampling time interval. The event rates for the present results are $N_{{events}}$ /blade-revolution = (figure 16 $a$ ) $1.2\times 10^6$ , (figure 16 $b$ ) $4.4 \times 10^4$ and (figure 16 $c$ ) $5.4 \times 10^3$ . The joint PDF of $C_p \lt -6.2$ in figure 16( $a$ ) shows a clear bimodal distribution, with low-pressure events more likely to occur in the trailing edge wake crossover region at $R\theta /c = 0.2$ and farther downstream at $R\theta /c = 0.4$ where high vortex stretching events were observed in figure 14( $b$ ). Low-pressure events are also seen to occur in the leakage vortex core and separation sheet structures forward of the blade trailing edge but at a much lower frequency than the vortex interaction regions in the blade wake. Furthermore, as the pressure threshold is decreased in figures 16( $b$ ) and 16( $c$ ) the probability density decays more significantly in the leakage vortex forward of the trailing edge than in the peak regions aft the blade trailing edge at $R\theta /c = 0.2$ and $0.4$ . This follows the experimental observations of Chesnakas & Jessup (Reference Chesnakas and Jessup2003) who observed cavitation inception at 50 % chord length aft, then, as the tunnel pressure was decreased, observed cavitation in secondary vortex structures closer to the trailing edge before the primary leakage vortex core cavitates as seen in figure 4( $a$ ). Joint PDFs for all three $C_p$ thresholds do indicate the presence of such low-pressure trailing edge vortices, but at a much lower frequency than in the downstream regions.

As discussed in reference to figure 13, the trailing edge wake is composed of highly unsteady small-scale vortical structures of differing signs of rotation. The statistics of extreme low-pressure events suggest that these small-scale vortical structures may intermittently align such that a low-pressure region does occurs in the trailing edge wake. However, the trailing edge low-pressure region is not a persistent feature that can be visualised in the pressure field at any given time instant, such as that shown in figure 12.

3.5. Mean flow field

This section examines the time-averaged flow field of the LES results on the finest discretisation (Grid 3) to provide a more in-depth explanation of the formation of instantaneous flow structures observed in the previous section. The average flow in inertial coordinates is computed in the frame of reference of the blade.

Figure 17. Isosurface of mean $Q$ -criterion coloured by mean pressure coefficient. Only the tip region $r/R \gt 0.85$ is shown for clarity.

The time-averaged vortex structures are examined in figure 17 with isosurface of mean $QR^2/U_\infty ^2 = 35$ . The time-averaged flow reveals a coherent vortex emanating from the blade trailing edge (B) in addition to the primary leakage vortex (A), as observed by Chesnakas & Jessup (Reference Chesnakas and Jessup2003) and Oweis et al. (Reference Oweis, Fry, Chesnakas, Jessup and Ceccio2006a ). Unlike the instantaneous flow, no vortex structures are observed in the separation sheet between the primary leakage vortex detachment and the blade trailing edge. The trailing edge vortex remains at a pressure higher than ambient ( $p_\infty$ ) until it crosses over top of the primary leakage vortex. The interaction between primary and trailing edge vortices in the crossover region is also associated with a reduction in the leakage vortex mean core pressure. The induced upstream counter-rotating vortex (C) is visible as a region of red to the left of the primary leakage vortex. Unlike what was seen instantaneously in figure 14( $a$ ), the time-averaged induced upstream vortex diffuses before it migrates towards the primary leakage vortex upper periphery. The $Q$ -criterion also identifies the attached boundary layer on the blade tip as a result of streamlines following the surface curvature.

Figure 18 shows slices of the mean flow field oriented parallel to the $x{-}r$ plane at the blade trailing edge and separated by $\unicode{x1D6E5} (R\theta /c) \approx 0.07$ . Figure 18(a) colours the slices by the azimuthal component of vorticity. The primary vortex core is visible as a region of strong positive azimuthal vorticity which originates near the midchord at the blade suction side tip. The primary vortex core is seen to separate aft of the midchord but upstream of the trailing edge and is offset in the axial direction. The separation sheet connecting the separated primary leakage vortex core is visible as another region of strong positive azimuthal vorticity. Aft of the trailing edge, the separation sheet rolls up into a second coherent vortex corotating with the primary leakage vortex. The pressure side blade wake is visible as a sheet of strong counter-rotating vorticity. The upstream-induced vortex is also visible as a region of strong counter-rotating vorticity located above the primary vortex core in the radial direction. The induced vortex originates from the rollup of the tip leakage flow duct boundary layer after the blade passes by, as discussed in § 3.3. The rollup can be explained by the removal of the strong adverse pressure gradient created by the blade’s passage which generates the reversed leakage flow in the tip gap. To more clearly delineate the boundaries of each vortex structure, isolines of the mean $Q$ -criterion  $=20$ and $220$ are also shown as black lines on each slice.

Figure 18. Slices through the time-averaged LES flow field computed on Grid 3 at $J=0.98$ coloured by ( $a$ ) azimuthal vorticity $\langle \omega _{\theta }\rangle$ , ( $b$ ) pressure coefficient $\langle C_{P} \rangle$ , ( $c$ ) mean-square pressure fluctuation $\langle p'p' \rangle$ , ( $d$ ) turbulent kinetic energy $k$ .

Comparison of figures 17 and 18( $a$ ) with figures 13 and 14( $a$ ) suggests the following differences between the instantaneous and time-averaged flow: (i) in the time average flow, the upstream induced counter-rotating vortex diffuses before migrating away from the duct wall to the primary leakage vortex periphery, and (ii) the coherent corotating trailing edge vortex in the time-average flow is instantaneously a mixture of corotating and counter-rotating small-scale rotation. The strongest mean vortices are both corotating while the instantaneous flow has both corotating and counter-rotating vortices, whose interaction (mutual stretching) is stronger than corotating vortex merger. The importance of these vortex dynamics to the cavitation inception behaviour is revealed by the analysis of the vortex stretching term in discussion of figure 14( $b$ ).

Figure 18(b) shows slices coloured by the mean pressure coefficient. A strong pressure gradient across the tip gap can be seen, which is driving the leakage flow. The location of the primary leakage vortex core seen in figure 18(a) coincides with a strong low pressure, as expected. Outside the vortex core, the suction side region of the blade is below $p_{\infty }$ forward of the trailing edge ( $s\lt 0$ ) but above $p_\infty$ aft of the trailing edge ( $s\gt 0$ ). This is relevant for cavitation prediction in the sense that an idealised vortex model (such as that employed by Chesnakas & Jessup (Reference Chesnakas and Jessup2003) and Oweis et al. (Reference Oweis, Fry, Chesnakas, Jessup and Ceccio2006a )) links vortex circulation to the pressure differential between the core and local ambient pressure. For the core to be at vapour pressure, the circulation of an idealised vortex would need to be larger aft of the trailing edge than it would forward nearer to the blade suction peak. The LES results show that vorticity is constantly being fed into the primary leakage vortex from the blade tip boundary layer and separation sheet, which causes the primary vortex to strengthen in circulation as $s$ increases. This tradeoff between the increasing ambient pressure and increasing circulation with increasing $s$ could potentially explain the lower event rates in the primary leakage vortex core near the trailing edge seen in figure 16.

Figure 18(c) displays the mean-square pressure fluctuation, which is a measurement of the unsteadiness of the vortex structures seen in the mean flow field in figure 18(a). Regions of high $\langle p'p' \rangle$ are visible in the primary separation sheet, the blade tip trailing edge region and on either side in the axial direction of the primary leakage vortex core moving aft. This is indicative of instantaneous vortices orbiting around the primary leakage vortex core, which is consistent with the interactions of small-scale vortex structures discussed earlier. More specifically, the observation of high mean-square pressure in the blade tip trailing edge provides evidence that the observed coherent trailing edge vortex observed in the experiments of Chesnakas & Jessup (Reference Chesnakas and Jessup2003) and Oweis et al. (Reference Oweis, Fry, Chesnakas, Jessup and Ceccio2006a ) is specifically a mean-flow phenomenon and that secondary vortices observed to cavitate during interaction with the primary leakage vortex may originate anywhere in the separation sheet connecting the blade tip boundary layer to the primary core. The large increase of $\langle p'p' \rangle$ on the upstream periphery of the vortex far aft of the blade trailing edge suggests that vortex core pressure minimum wanders in the axial direction. The regions of high $\langle p'p' \rangle$ correspond closely with the largest local $\partial \langle P \rangle /\partial x$ at $R \theta /c \gt 0.3$ .

Figure 18( $d$ ) shows the turbulent kinetic energy field. Forward of the blade trailing edge, $k$ is seen to peak in the upper periphery of the primary leakage vortex. As the primary leakage vortex moves away from the suction side blade tip, $k$ remains high throughout the separation sheet. Immediately behind the blade trailing edge, high turbulence levels are seen in the suction side blade wake and the primary leakage vortex outer periphery. Turbulence associated with the counter-rotating duct boundary layer rollup begins at the blade trailing edge and increases with additional distance aft the trailing edge. At the last slice, turbulence levels at the vortex periphery begin to dissipate and $k$ becomes highly concentrated in the primary leakage vortex core. High turbulence levels in the mean vortex core were also reported by Wu et al. (Reference Wu, Miorini, Tan and Katz2012) in the axial waterjet experiment.

Figure 19. Slices through the time-averaged LES flow field computed on Grid 3 at $J=0.98$ coloured by ( $a$ ) azimuthal vorticity $\langle \omega _\theta \rangle$ , ( $b$ ) azimuthal velocity $\langle u_{\theta } \rangle$ and ( $c$ ) axial velocity $\langle u_x \rangle$ .

3.5.1. Separation sheet skew

The mean flow also provides insight into the production of vortices aligned with the leakage flow, which were observed to wrap helically around the primary vortex in the preceding § 3.3. Figure 19( $a$ ) reveals that the entire separation sheet is a location of high axial vorticity which is not present in other regions of the flow. Figure 19 also shows the contours of the mean azimuthal velocity (figure 19 $b$ ) and axial velocity (figure 19 $c$ ) in the inertial frame. In addition to the primary shear in the axial flow ( ${\rm d}U_x/{\rm d}r$ ), there is also a secondary shear in the azimuthal flow ( ${\rm d}U_{\theta }/{\rm d}r$ ), representing a skewed shear layer. Skewed shear layers are a well-studied phenomenon and are known to produce vorticity aligned with the local flow (Lu & Lele Reference Lu and Lele1993; Fiedler et al. Reference Fiedler, Nayeri, Spieweg and Paschereit1998; Saric et al. Reference Saric, Reed and White2003; Wernz et al. Reference Wernz, Ringwald and Fasel2006; Meldi et al. Reference Meldi, Mariotti, Salvetti and Sagaut2020). Additionally, skewed shear layers have an instability associated with the inflection point of the velocity profile. Profiles of mean flow components across the separation sheet are shown in figure 20. Profiles are extracted in the radial direction at three locations upstream from the trailing edge $R\theta /c=-0.018,-0.065,-0.111$ and located halfway between the blade suction side and the primary leakage vortex core in the $x$ -direction. The vertical axis shows the radial coordinate normalised by the tip gap size $h$ and centred on the blade tip at $r=R$ . Above the separation sheet in the tip leakage flow, the axial velocity is strongly negative and the azimuthal velocity is moderately positive (moving opposite to the blade). The azimuthal velocity is moving with the blade just at the blade tip $r/R=1$ because fluid at this location originates in the blade boundary layer before separating off the suction side blade tip. Below the separation sheet, the azimuthal velocity is moderately negative (moving with the blade) while the axial velocity is strongly positive due to the radial circulation (or lift) produced by the blade passage. The relative strength of the flow components changes significantly with $s$ in the tip leakage flow, with profiles closer to the trailing edge having weaker azimuthal velocity and stronger axial velocity. Below the separation sheet, the dependence on $s$ is weaker. The axial velocity profile has a single inflection point at $r/R=0.998$ while the azimuthal velocity profile has two inflection points at $r/R=1.002$ and $0.995$ , respectively.

Figure 20. Mean velocity profiles across the separation sheet connecting the primary vortex core to blade tip boundary layer ( $a$ ) axial velocity $\langle u_x \rangle /U_\infty$ and ( $b$ ) azimuthal velocity $\langle u_{\theta } \rangle /U_\infty$ .

Figure 21. Reynolds stress tensor components from the LES Grid 3 solution at $J=0.98$ . Slices are parallel to the $x{-}r$ plane at the blade trailing edge and coloured by ( $a$ ) $\langle u_x^{\prime}u_r^{\prime} \rangle$ , ( $b$ ) $\langle u_r^{\prime}u_{\theta }^{\prime} \rangle$ , ( $c$ ) $\langle u_x^{\prime}u_{\theta } ^{\prime} \rangle$ and ( $d$ ) $\langle u_x^{\prime} u_x^{\prime} \rangle$ . The flow field quantities are normalised appropriately using $\rho$ , $U_{\infty }$ and $R$ .

The impact of the skew of the shear layer on turbulent structures is clearly visible in the Reynolds stress tensor components displayed in figure 21. Figure 21(a) displays slices of $\langle u_x^{\prime}u_r^{\prime} \rangle$ which demonstrates that in the separation sheet, the axial and radial velocity fluctuations are positively correlated. Meanwhile, figure 21(b) shows that the radial and azimuthal velocity fluctuations $\langle u_r^{\prime}u_\theta ^{\prime} \rangle$ are also positively correlated. Together, this provides evidence of instantaneous vortices that are corotating with the primary vortex but with a vortex core somewhat aligned with the axial direction, confirming the observations made in figure 13( $b$ ). Figure 21(c) shows that correlation of $\langle u_x^{\prime}u_{\theta } ^{\prime} \rangle$ flips sign on either side of the separation sheet. Above the separation sheet in the radial direction, fluctuations are positively correlated, while below the separation sheet, they are negatively correlated. However, correlations of $\langle u_x^{\prime}u_{\theta } ^{\prime} \rangle$ are approximately two times smaller than correlations of either $\langle u_x^{\prime}u_r^{\prime} \rangle$ or $\langle u_r^{\prime}u_\theta ^{\prime} \rangle$ in the separation sheet, indicating that instantaneous vortices in this region are unlikely to have a core aligned with the radial direction. This is the expected result given that the mean shear of the separation sheet is much larger in the radial direction. The regions of largest $\langle u_x'u_\theta '\rangle$ are instead found in the blade suction side boundary layer and wake where the dominant shear is in the axial direction.

Finally, figure 21(d) shows that the area between the induced upstream vortex and the edge of the primary leakage vortex is associated with a region of high turbulent fluctuation in the axial flow, $\langle u_x^{\prime}u_x^{\prime} \rangle$ . This indicates that this vortex is closely associated with the interaction of the tip leakage flow in the negative $x$ -direction and the incoming turbulent duct boundary layer in the positive $x$ -direction. A similar observation was made in the axial waterjet experiments of Wu et al. (Reference Wu, Miorini, Tan and Katz2012). This conclusion suggests that simulations performed with a steady or laminar inflow to the propulsor could predict a different vortex structure in the blade wake. Additionally, this suggests that the thickness of the duct boundary layer relative to the tip gap spacing could be an important scaling parameter that is non-trivial to achieve when comparing across experimental facilities.

In this section, mean flow features driving tip vortex structures for a ducted rotor at $J=0.98$ have been examined. The coherent trailing edge vortex described by Chesnakas & Jessup (Reference Chesnakas and Jessup2003) was revealed in the LES results to be a feature of large-time averages of the small-scale vortices in the region. Meanwhile, instability mechanisms not previously identified in the ducted propulsor flow were observed in the separation sheet connecting the blade tip boundary layer to the separated primary leakage vortex core. The importance of these instability mechanisms to the vortex structures of the separation sheet discussed in § 3.3 was shown in the off-diagonal terms of the Reynolds shear stress.