4.3.1. Turbulence anisotropy at the core of the tip vortices

The anisotropy of turbulence at the core of the tip vortices was analysed by considering the map of the invariants proposed by Lumley & Newman (Reference Lumley and Newman1977). They introduced the anisotropy tensor for turbulence, defined as

(4.2)\begin{equation} a_{ij} = \frac{\widehat{u'_i u'_j}}{2\hat{k}} - \frac{\delta_{ij}}{3},\quad i,j=1,2,3, \end{equation}

where $\widehat {u'_i u'_j}$ are the turbulent stresses (in the present case from phase-averaged statistics), $\hat {k}=0.5\widehat {u'_i u'_i}$ is the phase-averaged turbulent kinetic energy and $\delta _{ij}$ is the Kronecker delta. This tensor has three invariants. Actually, the first one is equal to zero, while the second and the third ones, $\mathcal {II}$ and $\mathcal {III}$, are utilized to characterize the level of anisotropy of the turbulence on the map shown in figure 12, depending on the relative importance of the different elements of the tensor. It should be noted that, for convenience, in the representation utilized by Lumley & Newman (Reference Lumley and Newman1977) the quantities $II$ and $III$ on the vertical and horizontal axes of figure 12 are actually proportional to the second and third invariants of the anisotropy tensor, $II=-2\mathcal {II}$ and $III=3\mathcal {III}$, respectively, as clarified in the later work by Lumley (Reference Lumley1979). The same definitions as in Lumley & Newman (Reference Lumley and Newman1977) were utilized in the present study. This approach to the analysis of the anisotropy of turbulence at the core of vortices was also recently adopted in the field of naval hydrodynamics in the works by Visonneau et al. (Reference Visonneau, Guilmineau and Rubino2018, Reference Visonneau, Guilmineau and Rubino2020).

Figure 12.

Turbulence anisotropy map by Lumley & Newman (Reference Lumley and Newman1977).

The map in figure 12 is bounded on the top side by the two-component turbulence, on the bottom left side by the axisymmetric ‘pancake-shaped’ turbulence, for which one component is smaller than the others, and on the bottom right side by the ‘cigar-shaped’ axisymmetric turbulence, for which one component is larger than the others. The top side of the map is represented by the equation $II=2/9+2III$, and the bottom side by the equation $3/2\ (4/3 |III|)^{2/3}$, describing both left and right sides of Lumley's map. In particular, the limiting case of the three-component isotropic turbulence, for which all elements of the anisotropy tensor are equal to 0, is characterized by values $II=0$ and $III=0$ (bottom vertex), the two-component isotropic turbulence by values $II=1/6$ and $III=-1/36$ (left vertex) and the one-component turbulence by values $II=2/3$ and $III=2/9$ (right vertex).

For each simulated condition the invariants of the anisotropy tensor were computed at the core of the tip vortices from phase-averaged statistics. In particular, at some selected streamwise coordinates the average of the turbulent stresses was computed within the core of the tip vortices. They were identified as local minima of pressure. Then, the Reynolds stresses were averaged within areas centred at the vortex core and having a radial extent equal to $0.002D$. These values were averaged further across all five tip vortices shed by the propeller. The results, in terms of anisotropy maps, are illustrated in figure 13, where only the lower region of the Lumley map is shown. This ‘zoomed-in’ representation of the map is chosen for clarity, since turbulence never moves further away from the isotropic state in the direction of one-dimensional turbulence.

Figure 13.

Turbulence anisotropy at the core of the tip vortices from phase-averaged statistics. Comparison across advance coefficients: (a) J0; (b) J1; (c) J2; (d) J3; (e) J4.

It should be noted that, although phase-averaged statistics are considered, the results are not dependent on the particular phase of the position of the propeller blades, since phase-averaged fields are characterized by the same symmetry as the propeller geometry and for different phases they are just rotated according to the particular phase of the propeller. Therefore, the results on the anisotropy of turbulence at each particular streamwise location are not affected by the choice of the phase angle, thanks to the symmetry of the open-water configuration of the propeller. In addition, averaging across all five tip vortices allows an increase in the size of the statistical sample, improving the time convergence of the statistics.

Figure 13 demonstrates that, just downstream of the propeller, at $z/D=0.5$, turbulence at the core of the tip vortices is close to isotropy. However, this is increasingly not the case for growing load conditions, for which turbulence anisotropy at $z/D=0.5$ gradually moves along the right-bottom boundary of the Lumley map. This behaviour was verified to be associated with a faster rise of the turbulent fluctuations of the radial velocity component, in comparison with the azimuthal and axial ones. This is the same trend verified across the streamwise evolution of the tip vortices. As their instability develops, the fluctuations of the radial velocity component grow at a faster rate, so turbulence at the core of the tip vortices moves away from isotropy along the right boundary of the Lumley map. This process is accelerated at higher loading conditions, that is, at lower values of the advance coefficient. Eventually, when the break-up of the tip vortices occurs, resulting in turbulent diffusion and more homogeneous values of Reynolds stresses, turbulence moves again towards a more isotropic state, as illustrated by the representation of its anisotropy in figure 13.

The streamwise evolution of turbulence anisotropy at the different working conditions is shown in better detail in figure 14, where for each case the three normal turbulent stresses along the radial, azimuthal and axial directions are reported. A number of results can be inferred from figure 14:

1. (i) Turbulent stresses are quickly growing for increasing rotational speeds (note the variation of the vertical scale across the five panels of figure 14).

2. (ii) The near wake is characterized by an increase of the level of anisotropy of turbulence at the core of the tip vortices, dominated by the fluctuations of the radial velocity component. In contrast, the lowest turbulent fluctuations affect the azimuthal velocity component.

3. (iii) In the near wake, turbulence grows as a result of the increasing instability of the tip vortices, affecting especially the radial velocity component.

4. (iv) Instability develops faster at lower advance coefficients, for which turbulence peaks closer to the propeller plane.

5. (v) The growing streamwise evolution across the near wake is followed by a decreasing trend of the turbulent stresses, affecting the radial and streamwise velocity components. In contrast, the turbulent fluctuations of the azimuthal velocity undergo an increase. These trends, associated with the break-up of the tip vortices and the resulting turbulence diffusion, lead to a more isotropic state of turbulence.

Figure 14.

Streamwise evolution of the phase-averaged normal, turbulent stresses at the core of the tip vortices in the radial ($\widehat {u'u'}$), azimuthal ($\widehat {v'v'}$) and axial ($\widehat {w'w'}$) directions, respectively. Comparison across advance coefficients: (a) J0; (b) J1; (c) J2; (d) J3; (e) J4. Note the variation of the vertical scale across panels.

A more global overview of the flow is reported in figures 15, 16 and 17, dealing with contours of normal, turbulent stresses from phase-averaged statistics, visualized on a meridian plane. For limitation of space, only the working conditions J0, J2 and J4 are considered, respectively. In the same figures, isolines of pressure coefficient were utilized again to isolate the minima of pressure occurring at the core of the tip (white isolines) and hub (magenta isolines) vortices. It should be noted that the colour scales for the Reynolds stresses are different across figures 15, 16 and 17, due to the substantial increase of turbulence levels across working conditions. This is also the case for the minima of pressure associated with the core of the tip and hub vortices at different advance coefficients. It should be also mentioned that the position of the wake structures, relative to the propeller, is not a function of the particular choice of the phase angle of its blades. As discussed above, the phase-averaged statistics are computed in synchronization with the rotation of the propeller, to capture the coherence of its wake. Therefore, the rotation of the phase angle of the propeller results in the same rotation of the phase angle of the phase-averaged statistics of the wake flow. In other words, a different phase from that considered in figures 15, 16 and 17 can be simply reconstructed by ‘cutting’ the same phase-averaged fields across a different meridian plane.

Figure 15.

Contours of phase-averaged normal, turbulent stresses at the working condition J0, scaled by $U_\infty ^2$: (a) radial, $\widehat {u'u'}$; (b) azimuthal, $\widehat {v'v'}$; (c) axial, $\widehat {w'w'}$. White and magenta isolines of pressure coefficients $\hat {c}_p=-0.4$ and $\hat {c}_p=-2.0$, respectively.

Figure 16.

Contours of phase-averaged normal, turbulent stresses at the working condition J2, scaled by $U_\infty ^2$: (a) radial, $\widehat {u'u'}$; (b) azimuthal, $\widehat {v'v'}$; (c) axial, $\widehat {w'w'}$. White and magenta isolines of pressure coefficients $\hat {c}_p=-1.2$ and $\hat {c}_p=-6.0$, respectively.

Figure 17.

Contours of phase-averaged normal, turbulent stresses at the working condition J4, scaled by $U_\infty ^2$: (a) radial, $\widehat {u'u'}$; (b) azimuthal, $\widehat {v'v'}$; (c) axial, $\widehat {w'w'}$. White and magenta isolines of pressure coefficients $\hat {c}_p=-2.0$ and $\hat {c}_p=-10$, respectively.

In figure 15 the development of instability phenomena for the tip vortices is reflected in the increase of the turbulent fluctuations and the diffusion of the peaks of turbulent stresses and pressure coefficient: the sharp maxima at the core of the tip vortices are replaced by wider areas of large turbulent stresses, which are the result of vortex meandering. Also figure 15 highlights the lead by the fluctuations of the radial velocity, affecting also wider areas of the outer boundary of the propeller wake as the instability of the tip vortices develops, if compared with the turbulent fluctuations of the other velocity components.

In figure 16 the anisotropy at turbulence is not substantially modified. Upstream of the break-up of the tip vortices and the diffusion of the turbulent stresses at the outer boundary of the propeller wake, the stresses associated with the fluctuations of radial velocity are higher. However, as demonstrated above by the statistics at the core of the tip vortices, the development of the wake is accelerated, due to the faster instability of the tip vortices: at the most downstream locations turbulence is closer to isotropy than in figure 15. These trends are reinforced in 17, dealing with the working condition corresponding to the lowest advance coefficient, that is, the highest propeller load. The signature of the tip vortices is lost more upstream, due to their faster break-up. As long as they remain coherent, the turbulent stresses are the highest in the radial direction and the lowest in the azimuthal direction. To summarize, for all cases of advance coefficient the development of instability phenomena by the tip vortices was found to be characterized by a shift from isotropic turbulence at their core towards a state of ‘cigar-shaped’ axisymmetric turbulence, dominated by the fluctuations of the radial velocity. After vortex break-up, turbulence experienced instead an opposite shift again towards isotropy.

4.3.2. Turbulence anisotropy at the core of the hub vortex

A similar analysis was conducted at the core of the hub vortex. Turbulent stresses were averaged at the core of the hub vortex by using the same criterion adopted for the tip vortices. Actually, for the hub vortex the dependence of the anisotropy of the turbulence on the load conditions was found to be reinforced, in comparison with the tip vortices. Figure 18(a) shows the anisotropy map for J0. Also in this case turbulence is initially close to isotropy. However, in contrast with the results for the tip vortices, as the hub vortex develops downstream, turbulence moves along the left boundary of the Lumley map, which indicates that it is approaching a two-component state. A similar but reinforced behaviour is observed at the conditions J1 and J2 (see figure 18b,c), with some exceptions in the former case at the most downstream locations, where turbulence tends to be dominated by a particular normal stress (the azimuthal one, as shown below), shifting to the right side of the anisotropy map. In contrast, at the most-loaded conditions turbulence remains close to isotropy at all streamwise coordinates, as shown in figure 18(d,e), which was found to be the result of a much quicker diffusion of the vortex core, as discussed in more detail below.

Figure 18.

Turbulence anisotropy at the core of the hub vortex from phase-averaged statistics. Comparison across advance coefficients: (a) J0; (b) J1; (c) J2; (d) J3; (e) J4.

Also for the hub vortex more details are provided by considering the streamwise evolution of the three normal turbulent stresses at the core of the hub vortex for each case of advance coefficient (figure 19). In figure 19(a), dealing with the working condition J0, turbulence levels at $z/D=0.5$ are quite homogeneous. However, the growing instability of the hub vortex leads to a substantial increase, affecting the radial and azimuthal normal stresses only, in agreement with the results observed in the anisotropy map of figure 18(a), indicating the development of a ‘pancake-shaped’ axisymmetric turbulence. In figure 19(b,c) the streamwise evolution is not substantially different from that in figure 19(a), but higher levels of turbulent stresses are achieved as the instability of the hub vortex develops. However, it should be also noted for the case J1 (figure 19b) that at the most streamwise locations a decline of $\widehat {u'u'}$ occurs, resulting in the dominance of the turbulent stress $\widehat {v'v'}$, reflected in the Lumley map of figure 18(b), where a shift of turbulence towards the right branch occurs. Important qualitative differences are instead observed in figure 19(d,e), dealing with the most-loaded conditions. Much higher values of turbulent stresses are achieved just downstream of the propeller at $z/D=0.5$, in comparison with the other, more lightly loaded conditions. Then, a dramatic drop occurs, as a result of the diffusion of turbulence at the core of the hub vortex, approaching an almost isotropic state already at about a diameter downstream of the propeller.

Figure 19.

Streamwise evolution of the phase-averaged normal, turbulent stresses at the core of the hub vortex in the radial ($\widehat {u'u'}$), azimuthal ($\widehat {v'v'}$) and axial ($\widehat {w'w'}$) directions, respectively. Comparison across advance coefficients: (a) J0; (b) J1; (c) J2; (d) J3; (e) J4. Note the variation of the vertical scale across panels.

Additional details can be inferred from the visualizations in figures 15, 16 and 17, where the core of the hub vortex is identified by means of the magenta isolines. They demonstrate that, at the working conditions J0 and J2, the core of the hub vortex remains coherent, characterized by sharp maxima of the turbulent stresses. In particular, in the near wake, higher values of radial and azimuthal turbulent stresses are achieved, in comparison with the axial ones. As the instability of the hub vortex develops, they spread across wider areas around the axis of the propeller wake, but keep a sharp peak at the wake axis. It should be noted that some details, dealing with the anisotropy of turbulence and illustrated through figure 19(a,c), are actually not distinguishable in figures 15 and 16. Their contours of the turbulent stresses were saturated for visibility of the maxima at the core of the tip vortices, which are lower than those at the wake axis. In contrast, it is interesting to notice that, in figure 17, dealing with the highest-loaded condition J4, the sharp minimum of pressure at the core of the hub vortex, identified by the magenta isoline just downstream of the propeller plane, experiences diffusion at a short distance away. This is due to the faster instability of the hub vortex, leading the maxima of the Reynolds stresses at the wake axis to experience a substantial drop, due to diffusion, and achieve quickly an almost isotropic state, as demonstrated in both figures 18(e) and 19(e).

4.4.1. Tip vortices

Figure 20 displays contours from phase-averaged statistics for all elements of the symmetric tensors $\hat {R}_{ij}^d$ and $-\hat {S}_{ij}$. The working condition J0 is considered. Here, $\hat {R}_{ij}^d$ is the deviatoric part of the tensor of the resolved Reynolds stresses

(4.3)\begin{equation} \hat{R}_{ij}^d = \widehat{u_i' u_j'} - \tfrac{2}{3} \hat{k} \delta_{ij},\quad i,j=1,2,3, \end{equation}

while $\hat {S}_{ij}$ is the resolved deformation tensor

(4.4)\begin{equation} \hat{S}_{ij} = \frac{1}{2} \left ( \frac{ \partial \hat{u}_i}{\partial x_j} + \frac{\partial \hat{u}_j}{\partial x_i} \right ),\quad i,j=1,2,3. \end{equation}

Figure 20.

Contours of (a) $\hat {R}_{ij}^d$ and (b) $-\hat {S}_{ij}$ from phase-averaged statistics at the working condition J0. Components of the tensors, scaled by $U_\infty ^2$ and $U_\infty /D$, respectively: (i) $rr$; (ii) $\vartheta \vartheta$; (iii) $zz$; (iv) $r\vartheta$; (v) $rz$; (vi) $\vartheta z$. White and black isolines of pressure coefficients $\hat {c}_p=-0.4$ and $\hat {c}_p=-2.0$, respectively.

In this section the tensors $\hat {R}_{ij}^d$ and $-\hat {S}_{ij}$ are compared, using the same criterion adopted in the work by Chow et al. (Reference Chow, Zilliac and Bradshaw1997a). A thorough overview is provided about the limitations of Boussinesq's hypothesis in reproducing the correct behaviour of the Reynolds stresses in the wake of marine propellers. These data could serve as a reference to the community for the correction of conventional, isotropic turbulence models. According to Boussinesq's hypothesis, the contours of the two tensors should display the same shape in the left and right panels of figure 20. It is evident that this is not the case at the core of the tip vortices, where the lobes of the contours display a different orientation. This is especially clear for the elements $rr$ (figure 20i), $zz$ (figure 20iii), $rz$ (figure 20v) and $\vartheta z$ (figure 20vi). This comparison provides confirmation that most RANS models are unsuitable to reproduce the behaviour of turbulence at the core of the tip vortices shed by propellers. This finding recommends the use of more sophisticated approaches of turbulence modelling, for instance based on the solution of the transport equation of the Reynolds stresses, relaxing Boussinesq's hypothesis of proportionality between the deviatoric part of the Reynolds stresses tensor and the deformation tensor. Although a similar result was found at the core of the hub vortex, the visualizations of figure 20 are not the best choice to capture it, so more details will be reported below on a plane orthogonal to the axis of the hub vortex. As expected, as the instability of the tip vortices develops, the Reynolds stresses experience an increase. In contrast, all elements of the deformation tensor from phase-averaged statistics undergo a decrease, due to the diffusion of the signature of the tip vortices in the mean velocity field, because of vortex meandering. In conventional turbulence modelling for RANS, this flow physics results in increasing values of the turbulent viscosity required to represent the Reynolds stresses, based on the gradients of the mean flow.

Although both Reynolds stresses and mean gradients at the core of the tip vortices undergo an increase at lower advance coefficients, results were found to be qualitatively similar at the other simulated working conditions, which is actually encouraging in terms of turbulence modelling. For limitation of space, this is illustrated only for the load conditions J2 and J4 in figures 21 and 22, where a similar visualization of the Reynolds stresses and deformation tensor is provided. Besides the increase of both of them (note the variation of the colour scales against those in figure 20), an earlier break-up of the coherence of the tip vortices is shown, whose signature in the phase-averaged fields is lost more quickly downstream of the propeller. As a result, the Reynolds stresses experience a faster streamwise growth, while the mean gradients decay more quickly than in figure 20, due to the diffusion of the signature of the tip vortices in the phase-averaged flow fields. In RANS modelling this is equivalent to increasing values of the turbulent viscosity to properly represent the Reynolds stresses as the propeller load grows.

Figure 21.

Contours of (a) $\hat {R}_{ij}^d$ and (b) $-\hat {S}_{ij}$ from phase-averaged statistics at the working condition J2. Components of the tensors, scaled by $U_\infty ^2$ and $U_\infty /D$, respectively: (i) $rr$; (ii) $\vartheta \vartheta$; (iii) $zz$; (iv) $r\vartheta$; (v) $rz$; (vi) $\vartheta z$. White and black isolines of pressure coefficients $\hat {c}_p=-1.2$ and $\hat {c}_p=-6.0$, respectively.

Figure 22.

Contours of (a) $\hat {R}_{ij}^d$ and (b) $-\hat {S}_{ij}$ from phase-averaged statistics at the working condition J4. Components of the tensors, scaled by $U_\infty ^2$ and $U_\infty /D$, respectively: (i) $rr$; (ii) $\vartheta \vartheta$; (iii) $zz$; (iv) $r\vartheta$; (v) $rz$; (vi) $\vartheta z$. White and black isolines of pressure coefficients $\hat {c}_p=-2.0$ and $\hat {c}_p=-10$, respectively.

To summarize the results above, Boussinesq's hypothesis was utilized to compute at the core of the tip vortices the turbulent viscosity that should be predicted by an isotropic RANS model, based on the phase-averaged Reynolds stresses and deformation tensor from the present LES computations

(4.5)\begin{equation} \nu_T ={-} \hat{R}_{ij}^d / 2 \hat{S}_{ij},\quad i,j=1,2,3. \end{equation}

Also in this case, the values at the core of the tip vortices were computed at each streamwise coordinate as averages across an area of radial extent equal to $0.002D$ and centred at the peak of negative pressure produced by the tip vortices. Boussinesq's hypothesis should result, at the same location, in the same value of $\nu _T$ across elements of the two tensors $\hat {R}_{ij}^d$ and $\hat {S}_{ij}$. This is obviously not the case, as demonstrated in figure 23, where the turbulent viscosity was scaled by using the molecular viscosity, $\nu$. It is also interesting to see that, in some cases, the ratio of (4.5) is characterized even by large negative values. Increasing load conditions result in higher values of turbulent viscosity and deviations from Boussinesq's hypothesis (note the variation of the vertical scale across the five panels of figure 23). This is especially the case when the process of instability of the tip vortices leads them to meandering and eventual break-up, which are characterized by a significant increase of $\nu _T$. These phenomena occur earlier at higher loads, for which the instability process of the tip vortices develops at a faster rate. Therefore, also the maxima of $\nu _T$ shift to upstream coordinates for lower values of the advance coefficient.

Figure 23.

Turbulent viscosity, computed as in (4.5), at the core of the tip vortices. Comparison across advance coefficients: (a) J0; (b) J1; (c) J2; (d) J3; (e) J4. Note the variation of the vertical scale across panels.

4.4.2. Hub vortex

Figure 24 provides a similar visualization of the elements of $\hat {R}_{ij}^d$ and $-\hat {S}_{ij}$ as figure 20, dealing with the case J0, but on the cross-section at $z/D=1.0$, which is orthogonal to the axis of the hub vortex. Also in figure 24 the data from LES highlight that the two tensors are not aligned, in contrast with Boussinesq's hypothesis for turbulence. For instance, at the core of the hub vortex, $-\hat {S}_{rr}$, $-\hat {S}_{\vartheta \vartheta }$ and $-\hat {S}_{\vartheta z}$ display a two-lobe structure that is missing for the corresponding Reynolds stresses $\hat {R}_{rr}^d$, $\hat {R}_{\vartheta \vartheta }^d$ and $\hat {R}_{\vartheta z}^d$, which are characterized instead by a strong axial peak (see the left and right panels of figure 24(i,ii,vi), respectively). The opposite phenomenon occurs for the elements $r\vartheta$ and $rz$ of the two tensors (left and right panels of figure 24(iv,v), respectively): while the deformation tensor is characterized by strong maxima at the wake axis, a two-lobe pattern is visualized in the corresponding contours relative to the Reynolds stresses, having a local minimum at the core of the hub vortex. Also, while $\hat {R}_{zz}^d$ in the left panel of figure 24(iii) achieves large values at the core of the hub vortex, characterized by a bimodal distribution with a negative peak at the wake axis and a positive one at slightly outer radial coordinates, the corresponding element of the deformation tensor in the right panel of figure 24(iii) is very small there. In the contours of figure 24 also the signature of one of the tip vortices is visible, confirming that also there the two tensors $\hat {R}_{ij}^d$ and $-\hat {S}_{ij}$ are not aligned. This is especially evident looking at the elements $rr$, $zz$ and $\vartheta z$ of the two tensors (figure 24i,iii,vi).

Figure 24.

Contours of (a) $\hat {R}_{ij}^d$ and (b) $-\hat {S}_{ij}$ from phase-averaged statistics at the working condition J0. Components of the tensors, scaled by $U_\infty ^2$ and $U_\infty /D$, respectively, at the streamwise coordinate $z/D=1.0$: (i) $rr$; (ii) $\vartheta \vartheta$; (iii) $zz$; (iv) $r\vartheta$; (v) $rz$; (vi) $\vartheta z$. White and black isolines of pressure coefficients $\hat {c}_p=-0.4$ and $\hat {c}_p=-2.0$, respectively. Dotted-dashed line encompassing the projection of the area swept by the propeller.

At the working condition J2 the comparison between $\hat {R}_{ij}^d$ and $-\hat {S}_{ij}$ is not substantially modified. This is shown in the contours of figure 25. At the core of the hub vortex local maxima are distinguishable for all diagonal elements $\hat {R}_{ii}^d$ as well as for $\hat {R}_{\vartheta z}^d$, while they are missing in the contours of the corresponding elements of $-\hat {S}_{ij}$ (figure 25i,ii,iii,vi). The opposite behaviour occurs for the elements $r \vartheta$ and $rz$ (figure 25iv,v). Once again, deviations between the orientations of the two tensors are also evident at the core of the tip vortex at outer radial coordinates.

Figure 25.

Contours of (a) $\hat {R}_{ij}^d$ and (b) $-\hat {S}_{ij}$ from phase-averaged statistics at the working condition J2. Components of the tensors, scaled by $U_\infty ^2$ and $U_\infty /D$, respectively, at the streamwise coordinate $z/D=1.0$: (i) $rr$; (ii) $\vartheta \vartheta$; (iii) $zz$; (iv) $r\vartheta$; (v) $rz$; (vi) $\vartheta z$. White and black isolines of pressure coefficients $\hat {c}_p=-1.2$ and $\hat {c}_p=-6.0$, respectively. Dotted-dashed line encompassing the projection of the area swept by the propeller.

Some more significant changes are distinguishable in figure 26, dealing with the heaviest-loaded condition J4, for which the instability of the hub vortex develops more quickly, affecting the shape of $\hat {R}_{ij}^d$. For instance, all elements of $\hat {R}_{ij}^d$ develop broader maxima and a bimodal distribution, characterized by a local minimum at the wake axis and a local peak at outer radial coordinates. Meanwhile, their comparison with $-\hat {S}_{ij}$ is still inconsistent with Boussinesq's hypothesis, with the exception of the elements $rz$: within the signature of the hub vortex the contours of $\hat {R}_{rz}^d$ and $-\hat {S}_{rz}$ in the left and right panels of figure 26(v) display a similar shape. Also $-\hat {S}_{r\vartheta }$ develops a bimodal distribution, but not correlating well with that for $\hat {R}_{r\vartheta }^d$ (see the left and right panels of figure 26iv), while all other elements of the tensor $-\hat {S}_{ij}$ are small within the hub vortex, where no local maxima are distinguishable, in contrast with the contours characterizing the deviatoric part of the tensor of the Reynolds stresses in the left panels of figure 26.

Figure 26.

Contours of (a) $\hat {R}_{ij}^d$ and (b) $-\hat {S}_{ij}$ from phase-averaged statistics at the working condition J4. Components of the tensors, scaled by $U_\infty ^2$ and $U_\infty /D$, respectively, at the streamwise coordinate $z/D=1.0$: (i) $rr$; (ii) $\vartheta \vartheta$; (iii) $zz$; (iv) $r\vartheta$; (v) $rz$; (vi) $\vartheta z$. White and black isolines of pressure coefficients $\hat {c}_p=-2.0$ and $\hat {c}_p=-10$, respectively. Dotted-dashed line encompassing the projection of the area swept by the propeller.

A more detailed comparison at the core of the hub vortex is reported in figure 27, using the same strategy adopted at the core of the tip vortices in figure 23. Figure 27(a) highlights that, especially for the elements $zz$ of $\hat {R}_{ij}^d$ and $-\hat {S}_{ij}$, Boussinesq's hypothesis is not accurate, resulting in very large and strongly variable values of the turbulent viscosity, much larger than those associated with all other elements of the two tensors: the values of $\hat {S}_{zz}$ are too small to be suitable to represent the corresponding Reynolds stresses $\hat {R}_{zz}^d$. This issue is actually diminished at lower advance coefficients, in contrast with the results observed at the core of the tip vortices, since for increasing loads the core of the hub vortex loses its coherence more quickly. However, it is still evident across all panels of figure 27 that the tensors $\hat {R}_{ij}^d$ and $-\hat {S}_{ij}$ are not aligned at the wake axis, resulting in a strong dispersion of the values of $\nu _T$ estimated from (4.5).

Figure 27.

Turbulent viscosity, computed as in (4.5), at the core of the hub vortex. Comparison across advance coefficients: (a) J0; (b) J1; (c) J2; (d) J3; (e) J4. Note the variation of the vertical scale across panels.

4.5.1. Tip vortices

This section provides evidence that the present computations were able to resolve most of turbulence, by comparing the resolved Reynolds stresses against the modelled ones. It is worth recalling that SGS modelling relies on the ability to resolve a wide range of scales, limiting modelling to the smallest scales only, which are more universal, homogeneous and isotropic. For them Boussinesq's hypothesis can be assumed to be more accurate. Moreover, all errors associated with this assumption affect a much narrower range of less energetic scales, compared with the conventional turbulence modelling adopted for RANS, with beneficial effects on the accuracy of the computations.

Figure 28 shows the ratios between all elements of the deviatoric parts of the tensors of the SGS stresses and resolved Reynolds stresses at the core of the tip vortices, indicated as $\hat {\tau }_{ij}^d$ and $\hat {R}_{ij}^d$, respectively, across cases of advance coefficient. All stresses were averaged within the core of the tip vortices, using the same criterion already discussed above. The ratios in figure 28 are usually well below $1\,\%$, with a few higher peaks, keeping within $10\,\%$. The only exception is the ratio $\hat {\tau }_{zz}^d / \hat {R}_{zz}^d$ for the case J2 in figure 28(c) at the streamwise location $z/D=0.5$. Interestingly, in all panels of figure 28 the streamwise trend is declining. This result indicates that the resolution of the computational grid is adequate to capture the downstream evolution of the tip vortices, as also demonstrated by its ability in reproducing their instability process, which was found to be in agreement with the one visualized in the physical experiments by Felli et al. (Reference Felli, Camussi and Di Felice2011): the contribution by SGS modelling is indeed diminishing towards downstream coordinates. It is also interesting to observe that the ratios in figure 28 are not a growing function of the load conditions, despite the increasing intensity of the tip vortices and Reynolds stresses resulting from their instability.

Figure 28.

Ratios between the deviatoric parts of the SGS and resolved Reynolds stresses tensors, $\hat {\tau }_{ij}^d$ and $\hat {R}_{ij}^d$, at the core of the tip vortices. Comparison across advance coefficients: (a) J0; (b) J1; (c) J2; (d) J3; (e) J4. Note the variation of the vertical scale across panels.

4.5.2. Hub vortex

The ratios $\hat {\tau }_{ij}^d / \hat {R}_{ij}^d$ are reported across cases of advance coefficient and streamwise coordinates at the core of the hub vortex in figure 29. Overall, values are even lower than those found at the core of the tip vortices. Also in figure 29 a streamwise decrease is distinguishable, with the highest peaks at the closest streamwise coordinates. Again, no obvious dependence on the working conditions and intensity of the hub vortex can be inferred, confirming that the overall approach remains suitable to represent the most energetic scales of the wake flow even moving towards higher-loaded conditions.

Figure 29.

Ratios between the deviatoric parts of the SGS and resolved Reynolds stresses tensors, $\hat {\tau }_{ij}^d$ and $\hat {R}_{ij}^d$, at the core of the hub vortex. Comparison across advance coefficients: (a) J0; (b) J1; (c) J2; (d) J3; (e) J4. Note the variation of the vertical scale across panels.