4.2.1 Statistics at the stern boundary layer

The boundary layer over the cylindrical mid-body is almost a canonical one, whose properties are extensively discussed in the literature, therefore our interest will concentrate towards the stern boundary layer, experiencing a strong adverse pressure gradient and very far from equilibrium conditions. A comparison between Reynolds numbers, concerning the growth of the boundary layer along the stern, is shown in figure 9. The boundary layer thickness is larger for $Re_{L}=1.2\times 10^{6}$ at all streamwise coordinates and the adverse pressure gradient causes substantial growth in both cases, as demonstrated by the slope of the evolution of $\unicode[STIX]{x1D6FF}_{1}$ and $\unicode[STIX]{x1D6FF}_{2}$. Note that at the beginning of the stern, $x/L\approx 0.75$, the boundary layer thickness experiences actually a slight decrease, since there the convex geometry of the hull produces a mild acceleration of the flow, as shown also by the distributions of pressure and skin-friction coefficients. The most significant boundary layer growth happens in the region $0.90<x/L<0.95$, where deceleration is the largest. A global view of the evolution of the boundary layer is shown in figure 10, via instantaneous snapshots of streamwise velocity in the $z=y$ plane between the stern appendages. It is clear that: (i) the numerical trip wire triggers a quick transition at the bow; (ii) the boundary layer thickens at the stern, due to the adverse pressure gradient; (iii) the thickness of the boundary layer at the stern sets the width of the wake, which is therefore narrower for $Re_{L}=12\times 10^{6}$. However, overall there is a weak dependence of the width of the near wake on the Reynolds number, as also reported in the experimental study by Jiménez et al. (Reference Jiménez, Hultmark and Smits2010a).

Figure 9.

Detail of the time-averaged displacement (a) and momentum (b) thickness distributions over the stern. Profiles extracted from the plane between the fins on the side away from the sail ($z=y$ and $y>0$).

Figure 10.

Instantaneous streamwise velocity in the plane between the stern appendages ($z=y$) from the LES computations at $Re_{L}=1.2\times 10^{6}$ (Posa & Balaras Reference Posa and Balaras2016a) (a) and $Re_{L}=12\times 10^{6}$ (b). Velocity scaled by $U_{\infty }$.

Figure 11.

Time-averaged profiles of tangential $u_{t}$ (a) and normal $u_{n}$ (b) velocity along the local direction normal to the stern surface at the streamwise locations $l_{1}$ ($x/L\approx 0.71$), $l_{2}$ ($x/L\approx 0.87$), $l_{3}$ ($x/L\approx 0.92$) and $l_{4}$ ($x/L\approx 0.96$). Comparison between $Re_{L}=1.2\times 10^{6}$ (dashed lines) and $Re_{L}=12\times 10^{6}$ (solid lines). Profiles extracted from the plane between the fins on the side away from the sail ($z=y$ and $y>0$). The variable $x_{n}$ stands for the local normal coordinate, while $U_{t}$ is the local component of the free-stream velocity tangential to the stern surface.

More details on the evolution of the turbulent boundary layer are reported in figure 11(a), where the time-averaged tangential velocity profiles are shown on the plane between the stern appendages, $z=y$, on the side opposite to the sail. They were extracted at the streamwise locations $x/L\approx 0.71$ ($l_{1}$), $x/L\approx 0.87$ ($l_{2}$), $x/L\approx 0.92$ ($l_{3}$) and $x/L\approx 0.96$ ($l_{4}$), which are at the end of the cylindrical mid-body, immediately upstream of the fins, at their trailing edge and downstream of them, respectively, as shown in figure 10. As expected from an equilibrium boundary layer, at $l_{1}$ the velocity profile is fuller at the higher Reynolds number, corresponding to smaller values of both displacement and momentum thickness. The shape of the velocity profiles is substantially modified along the stern, featuring a larger size of the outer layer, in comparison with the equilibrium region, due to the strong adverse pressure gradient. Note that at $l_{4}$ the velocity profiles experience, in the near-wall region, the additional influence of the junction flows from the appendages, bringing higher momentum in the inner layer. Differences between the two values of Reynolds number are actually more obvious in figure 11(b), where profiles of normal velocity component are reported. At $l_{1}$ the normal velocity is negligible. It is slightly negative, because of the mild acceleration at the edge between mid-body and stern. In contrast, at $l_{2}$ the normal velocity is a significant fraction of the tangential component: at both Reynolds numbers the boundary layer is growing quickly, with a substantial displacement of momentum away from the stern surface. Values of normal velocity become lower at the locations $l_{3}$ and $l_{4}$, but they remain quite significant. There, normal velocities are slightly higher at $Re_{L}=12\times 10^{6}$, especially in the near-wall region.

The effect of both adverse pressure gradient across the stern and junction vortices from both sail and fins is well represented in figure 12, where time-averaged contours of streamwise velocity are shown from downstream of the sail up to the tail of the DSub. For visualization purposes, contours were visualized enforcing a maximum threshold equal to $0.9U_{\infty }$. Results demonstrate for both Reynolds numbers that the junction vortices reorganize the momentum in the wake of both sail and fins, bringing higher momentum into the inner wake of the appendages and lower momentum towards their outer wake. It is also interesting to see that the effect of the junction vortices from the sail remains distinguishable over the whole length of the DSub, up to the fins, where the influence of their junction vortices becomes obviously dominant, producing significant azimuthal gradients, especially at the inner radii of the stern boundary layer. Also in figure 12 it is evident that, at the higher Reynolds number, the boundary layer is thinner, but the growth rate across the rear of the DSub is practically identical between the two considered cases. It is also very obvious at both Reynolds numbers that the boundary layer experiences, between the cylindrical mid-body and the stern, a mild acceleration, reducing locally its thickness.

Figure 12.

Time-averaged contours of streamwise velocity from the trailing edge of the sail up to the tail of the DSub. A maximum threshold equal to $0.9U_{\infty }$ was enforced for visibility purposes. Results at $Re_{L}=1.2\times 10^{6}$ (a) and $Re_{L}=12\times 10^{6}$ (b).

Figure 13.

Profiles of turbulent kinetic energy along the local direction normal to the stern surface in the plane between the fins on the side away from the sail ($z=y$ and $y>0$): (a) $x/L\approx 0.71$, (b) $x/L\approx 0.87$, (c) $x/L\approx 0.92$ and (d) $x/L\approx 0.96$.

In our earlier work (Posa & Balaras Reference Posa and Balaras2016a) the impact of the adverse pressure gradient on the turbulence distribution was found to be very significant at $Re_{L}=1.2\times 10^{6}$, affecting also wake features. In that case, we verified that, over the stern, the turbulence peak moves away from the inner layer, towards the outer layer. Such behaviour is actually associated with a decrease of the inner layer turbulence, while outer layer turbulence is almost unaffected. In the present study we observed a similar phenomenon at $Re_{L}=12\times 10^{6}$, although the generation of the outer turbulence peak is slower. This is due to lower values of turbulent kinetic energy in the outer layer, as shown in figure 13, where its profiles are represented along the direction normal to the wall again in the plane $z=y$ on the side away from the sail. Figure 13(a) refers to the streamwise location at $x/L\approx 0.71$ (end of the mid-body). Gradients of turbulent fluctuations between inner and outer layers are more significant at the higher Reynolds number. Values of inner layer turbulence are higher at $Re_{L}=12\times 10^{6}$ and vice versa in the outer layer. This is expected, due to the equilibrium nature of the turbulent boundary layer of the mid-body. Figure 13(b) shows the same distributions at the location $x/L\approx 0.87$ (upstream of the stern appendages). Values of turbulent kinetic energy in the outer layer are almost unchanged, compared to the upstream location, while the drop of inner layer turbulence is significant, especially at $Re_{L}=12\times 10^{6}$. However, turbulence is still higher in the inner layer. The same trend affects the profiles in figure 13(c) for the location $x/L\approx 0.92$ (streamwise coordinate of the trailing edge of the fins), but a local outer maximum of turbulent kinetic energy is already distinguishable at both Reynolds numbers. Further downstream in figure 13(d) ($x/L\approx 0.96$), the presence of an outer peak is much more obvious at $Re_{L}=1.2\times 10^{6}$. Again, outer layer turbulence is almost unaffected, while inner layer turbulence keeps decreasing. As a consequence of this trend, we will show that the bimodal distribution of turbulent fluctuations across the early streamwise development downstream of the DSub is stronger at the lower Reynolds number, because of the larger values of outer layer turbulence within the stern boundary layer, shed into the wake of the body.

4.2.2 Influence of stern appendages on flow features

Flow development along the stern is especially complex. It affects wake features, as extensively discussed in Posa & Balaras (Reference Posa and Balaras2016a), and the flow ingested by the propeller (Posa & Balaras Reference Posa and Balaras2016b, Reference Posa and Balaras2018), although propulsion is not the focus of the present work. Such complexity is tied to the adverse pressure gradient, but also to the presence of appendages and associated wake, tip vortices and junction vortices.

Figure 14.

Detail of the time-averaged displacement (a) and momentum (b) thickness distributions over the stern. Profiles extracted from the semi-plane $\unicode[STIX]{x1D6FF}$ at $11^{\circ }$ from the lower fin, whose projection is represented in figure 17. Thinner lines stand for the distributions over the semi-plane $\unicode[STIX]{x1D6FC}$ in figure 17, almost unaffected by the junction vortices.

The overall influence of the stern appendages on the development of the boundary layer at the rear of the DSub is shown in figure 14. There, displacement and momentum thicknesses are provided over the meridian semi-plane $\unicode[STIX]{x1D6FF}$, at $11^{\circ }$ from the lower fin, whose projection on a cross-stream section is shown in figure 17 reported below. For comparison, the evolution of the boundary layer thickness over the semi-plane $\unicode[STIX]{x1D6FC}$ (see figure 17 for its position) is also reported via thinner lines in the same figure 14 and taken as a reference of a boundary layer almost unaffected by the stern appendages. At the streamwise location corresponding to the leading edge of the fin ($x/L\approx 0.88$) for both Reynolds numbers both $\unicode[STIX]{x1D6FF}_{1}$ and $\unicode[STIX]{x1D6FF}_{2}$ display a couple of local maxima and minima. They are mainly originating from blockage phenomena and the influence of junction vortices there is negligible. The local maximum is indeed tied to lower values of tangential velocity, especially in the outer layer, because of the deceleration upstream of the appendage. Immediately downstream of the leading edge the flow experiences an acceleration, with a consequent sharp decrease of the boundary layer thickness. This is shown in figure 15(a), where the profiles of tangential velocity are provided for both Reynolds numbers over $\unicode[STIX]{x1D6FC}$ (thinner lines) and $\unicode[STIX]{x1D6FF}$ (thicker lines), demonstrating in the latter case an increase affecting the whole boundary layer thickness. The secondary effect of the junction vortices at this streamwise location, which is limited to the root of the stern appendages, is distinguishable at the inner coordinates, with a small local maximum of tangential velocity very close to the wall and a peak of negative normal velocity in figure 15(b). Thus, higher momentum fluid brought to the inner layer by the junction vortices adds to the decrease of the boundary layer thickness, but in this region of the stern boundary layer this effect is rather marginal and the acceleration near the leading edge of the fins is the main source of the increased tangential velocity. The boundary layer over $\unicode[STIX]{x1D6FF}$ accelerates rather uniformly across its whole thickness, with a more obvious increase of tangential velocity for smaller normal coordinates, since the azimuthal distance from the appendage grows moving away from the hull surface. As shown in figure 14, downstream of the leading edge of the fins the gap between distributions over $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FF}$ fades out. It grows again downstream of the appendages ($x/L>0.92$), due to the influence of the wake of the fin, causing additional momentum deficit. However, moving towards the tail of the DSub, the competing effect of the junction vortices, carrying higher momentum fluid, becomes gradually dominant, because of the momentum recovery within the wake of the fin. This is more evident for the displacement thickness, whose values over $\unicode[STIX]{x1D6FF}$ become lower than those over $\unicode[STIX]{x1D6FC}$ at the most downstream coordinates of the stern surface. This phenomenon is slightly more distinguishable at the higher Reynolds number, for which also the momentum thickness displays a similar trend very close to the tail.

Figure 15.

Time-averaged profiles of tangential (a) and normal (b) velocity along the local direction normal to the stern surface at the streamwise location $x/L=0.885$, immediately downstream of the leading edge of the fins. Thinner and thicker lines stand for the profiles over the semi-planes $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FF}$, respectively, represented in figure 17. Comparison between cases at $Re_{L}=1.2\times 10^{6}$ (dashed lines) and $Re_{L}=12\times 10^{6}$ (solid lines).

The effect of the junction flows on the shape of the boundary layer near the tail is discussed in more detail via figure 16, where time-averaged profiles of tangential velocity are reported at $x/L\approx 0.96$. In figure 16(a) the profiles over $\unicode[STIX]{x1D6FF}$ show an increase of tangential velocity at inner radii, compared to those over $\unicode[STIX]{x1D6FC}$, while in the outer region of the boundary layer they practically overlap each other. This behaviour is very similar between the two values of Reynolds number, albeit slightly stronger at $Re_{L}=12\times 10^{6}$. Also, at the present streamwise location the influence of the junction vortices extends further away along the azimuthal coordinate. For instance, figure 16(b) shows profiles of tangential velocity over the semi-plane $\unicode[STIX]{x1D6FE}$ (see its projection in figure 17 below), located at approximately $22^{\circ }$ from the lower fin, where values in the near-wall region differ from those over $\unicode[STIX]{x1D6FC}$. Note also that, being the impact of the junction vortices on velocity profiles limited to small normal coordinates, they affect the evolution of the displacement thickness more than that of the momentum thickness, as demonstrated in figure 14. Overall, figure 14 highlights that the growth of the boundary layer thickness is independent of Reynolds number for streamwise locations downstream of the cylindrical section of the DSub, revealing that it is dominated by the effect of the adverse pressure gradient and that the influence of the appendages, via blockage and root vortices, albeit very significant, is practically the same for the two simulated Reynolds numbers, at least within the stern boundary layer. We will show later that differences between the two cases will become more significant in the near wake of the DSub.

Figure 16.

Time-averaged profiles of tangential velocity along the local direction normal to the stern surface at the streamwise location $x/L=0.96$. In panels (a,b) thicker lines stand respectively for the profiles over the semi-planes $\unicode[STIX]{x1D6FF}$ and $\unicode[STIX]{x1D6FE}$, represented in figure 17. Thinner lines show the profiles over the semi-plane $\unicode[STIX]{x1D6FC}$. Comparison between cases at $Re_{L}=1.2\times 10^{6}$ (dashed lines) and $Re_{L}=12\times 10^{6}$ (solid lines).

Figure 17.

Contours of time-averaged streamwise velocity on the cross-section at the trailing edge of the stern appendages for $Re_{L}=1.2\times 10^{6}$ (a) and $Re_{L}=12\times 10^{6}$ (b). Also the projection of the streamlines of the secondary flows is shown, to emphasize the presence of junction vortices. Velocity scaled by $U_{\infty }$.

Figure 18.

Contours of turbulent kinetic energy on the cross-sections at $x/L\approx 0.88$ (a,f), $x/L\approx 0.90$ (b,g), $x/L\approx 0.92$ (c,h), $x/L\approx 0.94$ (d,i) and $x/L\approx 0.96$ (e,j) for $Re_{L}=1.2\times 10^{6}$ (a–e) and $Re_{L}=12\times 10^{6}$ (f–j). Values scaled using $U_{\infty }^{2}$. Also isolines of the second invariant of the velocity gradient tensor are shown to indicate the areas of highest rotation within the flow field.

Figure 19.

Profiles of turbulent kinetic energy at $x/L\approx 0.92$ along the local directions normal to the stern surface, represented in figure 17: profiles along $\unicode[STIX]{x1D6FD}$ (a), $\unicode[STIX]{x1D6FE}$ (b) and $\unicode[STIX]{x1D6FF}$ (c). For the profiles along the direction $\unicode[STIX]{x1D6FC}$ refer to figure 13(c).

More details on junction vortices are reported in figure 17, via the time-averaged field of streamwise velocity at the root of the fins on a cross-section at the trailing edge of the appendages. Also, the projection of the streamlines of the secondary flows is provided, to highlight the presence of junction vortices. Those streamlines were generated by removing from the velocity field the mean radial velocity across the stern boundary layer, in order to isolate the radial flows due to the junction vortices from those tied to the tapered geometry of the stern. Figure 17 shows that: (i) junction vortices affect substantially the streamlines at the root of the stern appendages; (ii) their size is nearly independent of Reynolds number; (iii) a complex pattern of secondary vortices is visible at both Reynolds numbers; (iv) an asymmetry relative to the horizontal plane is generated in both cases, due to the influence by the wake of the sail; (v) the core of the junction vortices is closer to the stern surface at the higher Reynolds number; since the junction vortices originate from the interaction of the stern boundary layer with the leading edge of the stern appendages, it is likely that the thinner boundary layer at $Re_{L}=12\times 10^{6}$ leads to junction flows closer to the stern surface.

Figure 20.

Isosurfaces of $Q$ ($QD^{2}/U_{\infty }^{2}=1$), coloured by streamwise velocity, from the time-averaged fields for $Re_{L}=1.2\times 10^{6}$ (a) and $Re_{L}=12\times 10^{6}$ (b). Velocity scaled by $U_{\infty }$.

Isolines of the second invariant of the velocity gradient tensor, $Q$, computed from the time-averaged flow fields, are shown in figure 18 at five cross-sections across the stern, to identify the cores of the junction vortices. They are reported on top of contours of turbulent kinetic energy. The impact of the junction vortices on turbulence at the root of the appendages is obvious and is rather similar between the two Reynolds numbers, at least in a qualitative sense. It is also interesting to see that the size of the areas of large rotation of the flow grows up to the trailing edge of the fins, at the streamwise location of panels (c,h), and then it starts decreasing, suggesting that the interaction of the junction vortices with the boundary layer over the appendages plays a role in their enhancement as they move downstream.

Figure 21.

Instantaneous contours of vorticity magnitude for $Re_{L}=1.2\times 10^{6}$ (a,c,e,g) and $Re_{L}=12\times 10^{6}$ (b,d,f,h) on the meridian planes $\unicode[STIX]{x1D6FC}$ (a,b), $\unicode[STIX]{x1D6FD}$ (c,d), $\unicode[STIX]{x1D6FE}$ (e,f) and $\unicode[STIX]{x1D6FF}$ (g,h). Detail over the stern. Values scaled by $U_{\infty }/D$.

Figure 22.

Contours of time-averaged streamwise velocity component on the cross-section at $x/L\approx 0.94$ for $Re_{L}=1.2\times 10^{6}$ (a) and $Re_{L}=12\times 10^{6}$ (b). Also the projection of the streamlines of the secondary flows is shown, to emphasize the presence of tip and junction vortices. Velocity scaled by $U_{\infty }$.

Overall, figure 18 confirms that the influence of Reynolds number on junction vortices is not dramatic in a qualitative sense. However, a more quantitative comparison allows us to provide a better insight into such influence. Values of circulation, $\unicode[STIX]{x1D6E4}$, were computed via integration of the streamwise vorticity across the areas defined by the isolines in figure 18 at the root of the stern appendages. We excluded from the integration the areas of large $Q$ associated with secondary radial flows over the surface of the appendages and in their wake, due to phenomena different from junction vortices (radial pressure gradients across the span of the fins). Values of circulation are reported in table 1. They are obviously larger at the higher Reynolds number. In both cases the evolution is non-monotonic, as suggested by the visualizations in figure 18, since they start decreasing downstream of the fins. Having verified that the effect of the junction vortices on statistics within the stern boundary layer is weakly dependent on Reynolds number, we can conjecture that a lower viscosity allows the generation of stronger junction vortices, but also reduces the propagation of their effect on the surrounding fluid within the stern boundary layer.

Table 1.

Circulation associated with the junction vortices found in figure 18, scaled by $DU_{\infty }$.

The effect of the junction vortices on turbulence distribution is shown in figure 19, where the profiles of turbulent kinetic energy along the normal to the stern surface are provided at the same streamwise position as in figure 17. They refer to the directions $\unicode[STIX]{x1D6FD}$, $\unicode[STIX]{x1D6FE}$ and $\unicode[STIX]{x1D6FF}$ represented in figure 17, while those along the direction $\unicode[STIX]{x1D6FC}$ were already reported above in figure 13(c). In figure 19(a) profiles along the direction $\unicode[STIX]{x1D6FD}$ are very similar to those along $\unicode[STIX]{x1D6FC}$, but junction vortices affect substantially turbulence distribution along the direction $\unicode[STIX]{x1D6FE}$ in figure 19(b). In the near-wall region turbulence is higher than away from the appendage, along the directions $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$, and an outer peak is not distinguishable at all for $Re_{L}=12\times 10^{6}$. Along $\unicode[STIX]{x1D6FF}$ (figure 19c) the footprint of the junction vortices is even more evident. Although the inner peak of turbulent kinetic energy is higher at the lower Reynolds number, near wall turbulence is again higher at $Re_{L}=12\times 10^{6}$, whereas away from the wall the opposite occurs. The overall influence of the junction vortices on turbulence distribution across azimuthal coordinates is similar at the two Reynolds numbers, although it is felt closer to the wall at $Re_{L}=12\times 10^{6}$.

A visualization of the junction vortices is provided via isosurfaces of $Q$ in figure 20, where also the presence of the tip vortices is very evident. Figure 20 is consistent with the results presented above: (i) the junction vortices originating at the leading edge of the stern appendages induce secondary vortices at closer azimuthal locations, eventually becoming the main vortices downstream; (ii) the junction vortices become stronger at the trailing edge of the fins, where the maximum value of circulation was observed at both Reynolds numbers; (iii) junction vortices are stronger at the higher Reynolds number, for which circulation was indeed found higher; (iv) the roll up between the two sides of each appendage and their tip generates well-distinguishable tip vortices.

Figure 21 displays instantaneous fields of vorticity magnitude at the rear of the DSub, providing a global overview of the evolution of the stern boundary layer. The meridian planes corresponding to the directions $\unicode[STIX]{x1D6FC}$, $\unicode[STIX]{x1D6FD}$, $\unicode[STIX]{x1D6FE}$ and $\unicode[STIX]{x1D6FF}$ are considered. In agreement with the statistics for turbulent kinetic energy, moving closer to the stern appendages, values of vorticity magnitude increase. This phenomenon is mainly due to the boundary layer over the fins and junction flows, thus it is already distinguishable in the region between the fins, affecting also the locations downstream of them. Over the planes $\unicode[STIX]{x1D6FE}$ and $\unicode[STIX]{x1D6FF}$, the closest to the plane of symmetry of the DSub, vorticity is higher on the upper side of each panel, due to the additional structures populating the wake of the sail. The above behaviours are similar between the two Reynolds numbers, although vorticity magnitude is higher at $Re_{L}=12\times 10^{6}$, with also higher values at the axis of the overall wake generated by the DSub. The size of the structures is obviously larger at $Re_{L}=1.2\times 10^{6}$, with the finer computational grid adopted at $Re_{L}=12\times 10^{6}$ proving to be able to resolve the smaller scales populating the stern boundary layer.

Figure 23.

Contours of turbulent kinetic energy for $Re_{L}=1.2\times 10^{6}$ (a–c) and $Re_{L}=12\times 10^{6}$ (d–f). Streamwise locations corresponding to $x/L\approx 0.92$ (a,d), $x/L\approx 0.94$ (b,e) and $x/L\approx 0.96$ (c,f). Values scaled by $U_{\infty }^{2}$.

The footprint of the junction vortices downstream of the stern appendages is shown in figure 22 for both Reynolds numbers at a location $x/L\approx 0.94$ in the wake of the fins. Also the presence of tip vortices is visible. They originate from the pressure gradient between the tip and the lateral surfaces of the fins, even at the 0 yaw angle considered in our computations. Again, asymmetries relative to the horizontal plane are due to the influence by the wake of the sail on the upper side. Compared to the upstream location considered above, the pattern in figure 22 is less complex, without significant secondary vortices associated with the primary junction vortices.

Junction vortices keep affecting turbulent kinetic energy fields in the wake of the appendages. In figure 23 locations corresponding to $x/L\approx 0.92$ (a,d), $x/L\approx 0.94$ (b,e) and $x/L\approx 0.96$ (c,f) are considered for both Reynolds numbers. In both cases junction vortices cause higher values of turbulent kinetic energy at the root of the fins, feeding additional turbulence into the stern boundary layer. In addition, downstream of the appendages, values of turbulent kinetic energy away from the wall are higher at the lower Reynolds number. The turbulence outer peak, coming from the stern boundary layer, is easily distinguishable for $Re_{L}=1.2\times 10^{6}$, whereas this is not the case at $Re_{L}=12\times 10^{6}$. This makes the role of the junction vortices in turbulence shed in the wake by the DSub relatively more significant at the higher Reynolds number. Also the wake of the sail affects turbulence fields: higher values of turbulent kinetic energy populate the upper region of each field for both Reynolds numbers. The comparison between Reynolds numbers highlights that also turbulence associated with the wake of the sail is higher at $Re_{L}=1.2\times 10^{6}$. Values are very high downstream of the fins. As expected, the width of their wake is larger at the lower Reynolds number. Note that such turbulence fades away quickly, keeping higher values at outer radii, due to the tip vortices originating from the stern appendages.

Figure 24.

Contours of production of turbulent kinetic energy for $Re_{L}=1.2\times 10^{6}$ (a–c) and $Re_{L}=12\times 10^{6}$ (d–f). Streamwise locations corresponding to $x/L\approx 0.92$ (a,d), $x/L\approx 0.94$ (b,e) and $x/L\approx 0.96$ (c,f). Values scaled by $U_{\infty }^{3}/D$.

Similarities between the computations carried out at the two Reynolds numbers are evident also in figure 24, where contours for the production of turbulent kinetic energy are shown. It is interesting to notice the complex pattern downstream of the root of the fins. Junction vortices enhance substantially the overall production, which is instead much smaller within the stern boundary layer, due to the dramatic reduction of the gradients over the rear of the DSub. This causes a drastic decrease of production, compared to that within an equilibrium boundary layer. Also in this case the role of the junction vortices is relatively more significant at the higher Reynolds number. As we are going to show, although production in the core of the junction vortices is slightly higher at $Re_{L}=1.2\times 10^{6}$, its contribution to overall production, relative to that in the stern boundary layer, is more significant at $Re_{L}=12\times 10^{6}$. Also in figure 24 we can see that the footprint of the wake of the fins, which is very evident in all panels, displays a more dramatic decreasing trend, in comparison with that at the inner radii, affected by the junction vortices. Moreover, in figure 24 the influence of the sail is weak. However, at both Reynolds numbers slightly higher values are visible at inner radii on the side of the sail, with instead lower values at outer radii. This is consistent with the results by Jiménez et al. (Reference Jiménez, Reynolds and Smits2010b), who reported a destructive effect by the wake of the sail on the shear stresses in the wake of the DSub, where they were slightly higher on the side opposite to the sail.

Figure 25.

Profiles of production of turbulent kinetic energy at $x/L\approx 0.92$ along the local directions normal to the stern surface, represented in figure 17: profiles along $\unicode[STIX]{x1D6FC}$ (a), $\unicode[STIX]{x1D6FD}$ (b), $\unicode[STIX]{x1D6FE}$ (c) and $\unicode[STIX]{x1D6FF}$ (d).

A more quantitative comparison is presented in figure 25, where the four panels show the profiles of production of turbulent kinetic energy along the directions $\unicode[STIX]{x1D6FC}$, $\unicode[STIX]{x1D6FD}$, $\unicode[STIX]{x1D6FE}$ and $\unicode[STIX]{x1D6FF}$, already defined in figure 17, at the streamwise location $x/L\approx 0.92$. It is worth noting that, away from the fins, the peak of production of turbulent kinetic energy remains in the inner layer, which is still the region with the largest velocity gradients, although a second lower peak is generated in the outer layer. Production in the near-wall region is higher at $Re_{L}=12\times 10^{6}$, whereas the opposite occurs away from the wall, consistent with the behaviour seen for the turbulent kinetic energy. Production experiences an increase in the vicinity of the lower fin. This is already visible in the outer region of the stern boundary layer along the direction $\unicode[STIX]{x1D6FE}$ in panel (c), where production is enhanced, compared to that in panels (a,b). However, the effect of the junction flows becomes more evident along $\unicode[STIX]{x1D6FF}$ in panel (d), with again higher values near the wall at the higher Reynolds number and away from the wall at the lower Reynolds number, since junction vortices keep closer to the stern surface in the former case. It is also interesting to notice from figure 25(d) that, for both Reynolds numbers, the near-wall trend is actually the result of the overlap of multiple local maxima, as also shown in figure 24.

Figure 26.

Azimuthal profiles of streamwise velocity (a,c,e) and turbulent kinetic energy (b,d,f) for $Re_{L}=1.2\times 10^{6}$ (dashed lines) and $Re_{L}=12\times 10^{6}$ (solid lines) at the streamwise location $x/L\approx 0.94$. Distributions at $r/D=0.15$ (a,b), $r/D=0.25$ (c,d) and$r/D=0.35$ (e,f).

The contours above highlight that the impact of the junction vortices on first-order and second-order statistics within the stern boundary layer is not substantially dependent on Reynolds number, at least for a change of an order of magnitude, as in the present case. This is demonstrated more quantitatively by figure 26, where azimuthal profiles of streamwise velocity component (left) and turbulent kinetic energy (right) are shown at $x/L\approx 0.94$. Results at additional streamwise locations between the fins and downstream of them were not reported here due to limitation of space, but we verified that they agree with the conclusions below. At the radial coordinate $r/D=0.15$ the momentum gain associated with the junction vortices generates maxima of streamwise velocity at the azimuthal coordinates corresponding to the locations of the fins (figure 26a). The low momentum fluid in the wake of the appendages is carried away from those azimuthal locations by the junction vortices, producing this way the local minima seen at both sides of each peak of streamwise velocity. The effect of the junction vortices generated by the sail is very obvious even at the tail of the DSub for both Reynolds numbers, since the maximum at $\unicode[STIX]{x1D717}=180^{\circ }$ is significantly higher than those in the wake of the fins not aligned with the sail. The azimuthal extent of the region affected by the junction vortices is similar for the two Reynolds numbers, as well as the momentum gain directly downstream of the fins, relative to the streamwise velocity within the stern boundary layer. This momentum gain is only slightly higher at $Re_{L}=12\times 10^{6}$. In figure 26(b) values of turbulent kinetic energy at the couples of maxima associated with the junction vortices are very close between Reynolds numbers, in contrast with the lower values of outer layer turbulence for the higher Reynolds number at the azimuthal coordinates away from the fins. Also in the azimuthal distributions of turbulent kinetic energy the footprint of the junction vortices from the sail is still significant, being the source of the higher values observed around $\unicode[STIX]{x1D717}=180^{\circ }$. Moving away from the surface of the stern at $r/D=0.25$ (figure 26c) the influence of the junction vortices is weaker, but it remains easily distinguishable. Minima of streamwise velocity occur directly downstream of the fins, because of the momentum deficit in their wake, but the maxima at both edges of each wake confirm the significant role of the junction vortices in readjusting the distribution of momentum downstream of the appendages. Also at this radial location such influence is not obviously dependent on the Reynolds number, although in this panel it is slightly stronger at $Re_{L}=1.2\times 10^{6}$, with the exception of the region aligned with the wake of the sail. In figure 26(d) the highest values of turbulent kinetic energy are associated with the wake of the fins, but local maxima/minima still populate the edge of those wakes. In agreement with the results for the streamwise velocity, except for the area around $\unicode[STIX]{x1D717}=180^{\circ }$, the distribution away from the wake of the fins is more uniform at the higher Reynolds number. As discussed above, looking at the normal profiles of turbulent kinetic energy, at $Re_{L}=12\times 10^{6}$ the influence of the junction vortices is felt slightly closer to the surface of the stern, thus it fades out more quickly moving away to larger radial coordinates. Further away from the hull surface, at $r/D=0.35$ (figure 26e,f), the footprint of the junction vortices from the stern appendages is missing in both momentum and turbulence distributions. Instead, that of the junction vortices from the sail is still a source of small local maxima of streamwise velocity and turbulent kinetic energy at the edges of the wake centred at $\unicode[STIX]{x1D717}=180^{\circ }$, although the most evident flow features are obviously the momentum deficit in the wake of the fins and the turbulence peak within the shear layer shed from their trailing edge.

Figure 27.

Instantaneous contours of streamwise velocity for $Re_{L}=1.2\times 10^{6}$ (a–c) and $Re_{L}=12\times 10^{6}$ (d–f). Streamwise locations corresponding to $x/L\approx 0.92$ (a,d), $x/L\approx 0.94$ (b,e) and $x/L\approx 0.96$ (c,f). Values scaled by $U_{\infty }$.

The increase of momentum directly downstream of the fins in the near-wall region is shown in figure 27, where instantaneous contours of streamwise velocity are reported at the same streamwise locations considered in figure 23. The momentum deficit at inner radii is higher away from the appendages, with azimuthal gradients increasing moving from the left to the right sides of figure 27. The opposite occurs at outer radii, where momentum deficit is higher in the wake of the fins, but the wake recovery across the panels of figure 27 makes the azimuthal distribution of velocity more uniform moving away from the stern appendages. Also, in agreement with the discussion above, although the thickness of the stern boundary layer is slightly dependent on the Reynolds number, its size is dominated by the effect of the adverse pressure gradient. The difference between the two Reynolds number cases is indeed small, compared to the overall boundary layer thickness at the tail of the DSub.

4.2.3 Near-wake development

Figure 28.

(a) Time-averaged contours of streamwise velocity at the tail of the DSub for $Re_{L}=1.2\times 10^{6}$ (left) and $Re_{L}=12\times 10^{6}$ (right); (b) profiles along the direction $\unicode[STIX]{x1D709}$ shown in (a); (c) profiles along the directions $\unicode[STIX]{x1D702}$ shown in (a). Velocity scaled by $U_{\infty }$.

In this section we will compare the near-wake development for the two simulated Reynolds numbers. Figure 28(a) shows time-averaged contours of streamwise velocity as well as isolines for both $Re_{L}=1.2\times 10^{6}$ (left) and $Re_{L}=12\times 10^{6}$ (right) at the tail of the DSub. With no appendages isolines would be circumferential. Their deformation is caused by the wake of the appendages and by the junction and tip flows originating from them. Inner radii show the effect of the junction vortices. The deformation of the isolines there is due to the higher momentum fluid that junction vortices bring in the wake of the fins, with lower momentum fluid away from them. Such deformation is more evident at $Re_{L}=12\times 10^{6}$. Streamwise velocity profiles were extracted along the directions $\unicode[STIX]{x1D709}$ and $\unicode[STIX]{x1D702}$ and compared between the two Reynolds numbers in figure 28(b) and figure 28(c), respectively. In figure 28(b) the comparison along $\unicode[STIX]{x1D709}$ shows: (i) a higher momentum deficit at the lower Reynolds number; (ii) a similar momentum deficit due to tip flows ($r/D\approx \pm 0.5$); (iii) a similar footprint of the wake of the sail for the two Reynolds numbers ($r/D<-0.5$). Profiles in figure 28(c) along $\unicode[STIX]{x1D702}$ are not affected by the wake of the sail and the tip flows from the fins and the effect of the junction vortices from the stern appendages can be isolated. Compared to figure 28(b), momentum deficit at inner radii is higher, which is due to the lower momentum fluid that junction vortices carry away from the wake of the fins, from the azimuthal location $\unicode[STIX]{x1D709}$ towards the azimuthal location $\unicode[STIX]{x1D702}$. This is especially evident at the higher Reynolds number, with local minima at radial coordinates $r/D\approx \pm 0.1$. In contrast, at outer radii momentum deficit is lower, compared to the profiles along $\unicode[STIX]{x1D709}$, being $\unicode[STIX]{x1D702}$ away from the wake of both sail and fins.

Figure 29.

(a) Contours of turbulent kinetic energy at the tail of the DSub for $Re_{L}=1.2\times 10^{6}$ (left) and $Re_{L}=12\times 10^{6}$ (right); (b) profiles along the direction $\unicode[STIX]{x1D709}$ shown in (a); (c) profiles along the directions $\unicode[STIX]{x1D702}$ shown in (a). Values scaled by $U_{\infty }^{2}$.

Results for turbulent kinetic energy at the tail are provided in figure 29. Figure 29(a) shows higher values at outer radii for $Re_{L}=1.2\times 10^{6}$, which displays a more obvious outer maximum, in agreement with the discussion on boundary layer development in § 4.2.1. We have found, looking at the boundary layer over the stern, that the near-wall turbulence fades away dramatically at both Reynolds numbers and the inner peak is not visible anymore upstream of the tail (see figure 13d). However, due to turbulence associated with junction vortices, in figure 29(a) a local maximum of turbulent kinetic energy is produced at inner radii, even along the directions $\unicode[STIX]{x1D702}$, especially at the higher Reynolds number. We recall that the azimuthal extent of the junction vortices, relative to the stern, grows moving downstream, due to the tapered geometry of the aft of the DSub. Therefore they are able to affect also azimuthal coordinates away from the appendages. Figure 29(b) and figure 29(c) allow more quantitative considerations. Profiles along $\unicode[STIX]{x1D709}$ show two maxima on each side. The outer ones are due to the tip flows from the stern appendages, while the inner ones to the junction flows. This azimuthal coordinate is strongly affected by the wake of the appendages, especially the fins. Therefore, turbulence is higher at the lower Reynolds number. Asymmetry is due to the wake of the sail, causing higher turbulence on the left side of figure 29(b). Profiles along $\unicode[STIX]{x1D702}$ in figure 29(c) are substantially different. At outer radii values are lower than in figure 29(b), since the influence of the wake of the appendages is missing. Also in this case two maxima are visible on both sides, but the source of the outer ones is different than in figure 29(b). They originate from the thick stern boundary layer, which developed an outer peak of turbulent kinetic energy. As discussed in § 4.2.1, that peak is larger at the lower Reynolds number. In contrast, the inner maxima, associated with the junction vortices, are substantially higher at $Re_{L}=12\times 10^{6}$. Therefore, compared to upstream locations, results at the tail of the DSub indicate that junction vortices from the stern appendages develop gradually a more significant role in both first- and second-order statistics at the higher Reynolds number, with a stronger impact on the near wake of the DSub, albeit over most of the stern boundary layer their footprint is rather similar between $Re_{L}=1.2\times 10^{6}$ and $Re_{L}=12\times 10^{6}$. The discussion in § 4.2.2 pointed indeed to the higher values of circulation within the junction vortices in the latter case, consistent with their growing influence as the wake of the DSub develops downstream.

Figure 30.

Contours of turbulent kinetic energy on the horizontal meridian plane ($y=0$) for $Re_{L}=1.2\times 10^{6}$ (a) and $Re_{L}=12\times 10^{6}$ (b). Values scaled by $U_{\infty }^{2}$.

Figure 31.

Turbulent kinetic energy in the wake of the appended DSub for $Re_{L}=1.2\times 10^{6}$ (a) and $Re_{L}=12\times 10^{6}$ (b) on 6 cross-sections between $x/D=0$ and $x/D=5$. Values scaled by $U_{\infty }^{2}$.

The discussion above pointed to the fact that, at the tail, the outer maxima of turbulent fluctuations, associated with the stern boundary layer, are larger at the lower Reynolds number. However, the evolution downstream for $Re_{L}=1.2\times 10^{6}$ and $Re_{L}=12\times 10^{6}$ is different, making the dual peak in the wake slightly stronger at the higher Reynolds number, in agreement with the results reported for the unappended DSub by Jiménez et al. (Reference Jiménez, Hultmark and Smits2010a). Figure 30 shows a comparison of the fields of turbulent kinetic energy in the horizontal plane ($y=0$). The boundary layer over the hull is populated by higher values of turbulent kinetic energy for $Re_{L}=1.2\times 10^{6}$. Nonetheless, the bimodal distribution in the wake is similar between the two cases, also in quantitative terms, with even slightly higher values for $Re_{L}=12\times 10^{6}$. It is also worth noting that, in both cases, the maxima of turbulent kinetic energy experience a slight increase moving downstream, due to azimuthal readjustment of the wake, as we will discuss more in detail later. Turbulence at the trailing edge of the fins is substantially higher than that shed from the stern boundary layer, especially at outer radii, but it fades out very quickly downstream and boundary layer turbulence already dominates the wake signature at only few diameters away from the tail of the DSub.

Figure 32.

(a) Contours of turbulent kinetic energy 1 diameter downstream of the DSub for $Re_{L}=1.2\times 10^{6}$ (left) and $Re_{L}=12\times 10^{6}$ (right); (b) profiles along the direction $\unicode[STIX]{x1D709}$ shown in (a); (c) profiles along the direction $\unicode[STIX]{x1D702}$ shown in (a). Values scaled by $U_{\infty }^{2}$.

Figure 33.

(a) Contours of turbulent kinetic energy 5 diameters downstream of the DSub for $Re_{L}=1.2\times 10^{6}$ (left) and $Re_{L}=12\times 10^{6}$ (right); (b) profiles along the direction $\unicode[STIX]{x1D709}$ shown in (a); (c) profiles along the direction $\unicode[STIX]{x1D702}$ shown in (a). Values scaled by $U_{\infty }^{2}$.

The azimuthal readjustment of turbulent kinetic energy is well represented in figure 31, where 6 cross-sections are shown between the streamwise locations at $x/D=0$ and $x/D=5$. At the tail distributions are obviously not axisymmetric, with the footprint of the stern appendages easily distinguishable, besides that of the wake of the sail. Values are higher at the lower Reynolds number. In the near-wake turbulence originating from the fins decreases very quickly moving downstream and a stronger bimodal distribution is visible in the planes away from the appendages, with instead lower values at azimuthal coordinates directly aligned with them. However, moving away from the DSub, the turbulence distribution becomes more symmetric, increasing in the planes through the fins, as pointed in the discussion dealing with figure 30. Note also that, at downstream locations, values of turbulent kinetic energy are actually higher at $Re_{L}=12\times 10^{6}$. A more quantitative evidence of this trend is provided in figure 32. We have shown above that at the tail maxima of turbulent stresses, associated with the stern boundary layer, are higher at the lower Reynolds number. This is not the case anymore already at $1D$ downstream. Figure 32(a) displays a side-by-side comparison between Reynolds numbers. Maxima away from the axis are already very similar. It is also confirmed that in the DSub wake turbulence due to the junction vortices is higher at the higher Reynolds number, as visible at inner radii at azimuthal coordinates closer to the planes of the appendages ($y=0$ and $z=0$). Profiles in figure 32(b), extracted along the direction $\unicode[STIX]{x1D709}$, show for both Reynolds numbers a substantial decrease of the outer maxima, associated with tip flows from the stern appendages, compared to figure 29(b). In contrast, the inner maxima, mainly tied to the junction flows from the root of the fins, are almost unchanged. Furthermore, at this streamwise location they are already slightly higher at the higher Reynolds number, confirming that their footprint in the near wake becomes more significant for $Re_{L}=12\times 10^{6}$. Also the profiles along $\unicode[STIX]{x1D702}$ in figure 32(c) show interesting results. The outer peaks, associated with turbulence coming from the stern boundary layer, become very similar at this streamwise location, in contrast with the results reported in figure 29(c). The inner maxima, tied to the junction vortices, are again higher for the case at $Re_{L}=12\times 10^{6}$, while they are already practically missing at $Re_{L}=1.2\times 10^{6}$.

As seen in figure 31, at downstream locations in the wake, the turbulence distribution becomes more axisymmetric, with slightly higher values of turbulent kinetic energy at the higher Reynolds number, especially at azimuthal coordinates aligned with the stern appendages, which is likely a memory of the stronger footprint of the junction vortices. More details are reported in figure 33, via the comparison between the two Reynolds numbers at 5 diameters from the tail, where figure 33(a) deals with the most downstream section in both panels of figure 31. In figure 33(b) the maxima along the direction $\unicode[STIX]{x1D709}$ are indeed higher at $Re_{L}=12\times 10^{6}$, with still some memory of the wake of the fins at outer radii for both Reynolds numbers. Along the direction $\unicode[STIX]{x1D702}$ (figure 33c) profiles are very similar between $Re_{L}=1.2\times 10^{6}$ and $Re_{L}=12\times 10^{6}$. This result is in agreement with the conclusions by Jiménez et al. (Reference Jiménez, Hultmark and Smits2010a), dealing with the development of the wake of the unappended DSub, who reported that the bimodal distribution of turbulence in the near wake is weakly dependent on Reynolds number, with only slightly lower fluctuations at the lowest value they investigated, equal to $Re_{L}=1.1\times 10^{6}$. We also verified that less coherent structures populate the wake of the DSub at the lower Reynolds number. This is shown in figure 34, where the $Q$-criterion was utilized. The bimodal distribution seen for the turbulent kinetic energy fields is very well distinguishable also in terms of wake coherence, but at $Re_{L}=1.2\times 10^{6}$ the structures originating from the stern boundary layer in figure 34 extend up to smaller distances downstream. At both Reynolds numbers the signature of the wake shed from the stern appendages is already missing few diameters downstream of the tail of the DSub, which is also consistent with the results reported for the statistics in the wake.

Figure 34.

Instantaneous visualization of coherent structures using the $Q$-criterion at $Re_{L}=1.2\times 10^{6}$ (a) and $Re_{L}=12\times 10^{6}$ (b) ($QD^{2}/U_{\infty }^{2}=80$). Coloured by streamwise velocity, scaled by $U_{\infty }$.