3.2.1. Sectional analysis at $\textit {Re}=1500$

The convections of radial vorticity due to tangential and vertical flow ($C_t$ and $C_y$) are observed after the onset of wing rotation, and their intensities increase until the steady state ($\lambda =4$). As shown in figure 6(a), the downstream convection of negative radial vorticity ($C_t^{(+)}$ towards $C_t^{(-)}$) behind the leading edge feeds radial vorticity into the LEV. In addition, the downward convection of negative radial vorticity ($C_y^{(+)}$ towards $C_y^{(-)}$) behind the wing can further transport the LEV vorticity towards the trailing edge (figure 6b). Together, these convections prevent the LEV from continuously growing into the downstream direction and lead to its chord-aligned expansion. The downward convection of radial vorticity is also relevant to the tip-vortex-induced downwash effect (Birch & Dickinson Reference Birch and Dickinson2001). For brevity, only steady-state $C_t$ and $C_y$ at $\lambda =4$ are shown here since the only quantitative difference is observed during their formations towards the steady state. Note that these two vorticity convections become incoherent around the tip, which introduces strong unsteadiness into the evaluation of local LEV.

Figure 6.

Convection of radial vorticity at $\textit {Re}=1500$: (a) steady-state tangential convection ($C_t$), (b)  steady-state vertical convection ($C_y$), and (c–f)  evolution of spanwise convection ($C_r$). The radial velocity ($u_r$), pressure gradient (${F}_{pre}$) and Coriolis acceleration (${F}_{cor}$) on two representative cylindrical slices A and B (0.25b and 0.5b) are shown in panels i to iv, with the vortical region circled by ${Q}=1$. Here, $\hat {C}_{r}$ is the integral of $C_r$ within the LEV region.

While the downstream and downward convection of radial vorticity might explain the primary source and drain of LEV vorticity, the saturation of LEV intensity should also be attributed to other vorticity transport mechanisms that continuously remove the radial vorticity of the LEV, leading to a dynamic equilibrium. One of the possible contributors is the radial (spanwise) convection of radial vorticity ($C_r$), which is shown in figure 6(c–f) for the LEV formation period. Two representative slices A and B are selected at 0.25b and 0.5b, respectively, and a positive $C_r$ ($C_r^{(+)}$) indicates tip-ward convection of negative radial vorticity. It is found that a strong $C_r^{(+)}$ transports the radial vorticity towards the tip during the entire LEV formation period. The region of this strong $C_r^{(+)}$ is initially extended beyond the mid-span (red tube-shaped region, figure 6c, A-i and B-i) but tends to be confined within the mid-span afterwards (figure 6d–f, A-i and B-i). Moreover, a significant C $_r^{(-)}$ region is observed where the LEV contacts the wing surface (especially around mid-span), indicating an enhancement of radial vorticity (blue region in figure 6c, B-i). Therefore, the integral of total $C_r$ at the steady state is characterized by a reduction of radial vorticity near the wing root and a local enhancement around the mid-span ($\lambda >4$). The $C_r^{(+)}$ in the inboard region, e.g. 0.25b, is sustained under the steady state (figure 6d–f, A-i). Note that the chordwise $C_r^{(+)}$ region at a specific spanwise position is mostly located in the downstream boundary of the LEV (behind the local $C_r^{(-)}$ region). The spatial complexities of $C_r^{(+)}$ and $C_r^{(-)}$ in the LEV region are further elaborated in § 3.2.3. Note that no significant tip-ward vorticity convection is observed in the unsteady LEV region, i.e. outboard region beyond the mid-span, during the entire evolution.

To explain the behaviours of tip-ward vorticity convection, the evolution of spanwise flow, ${F}_{pre}$ and ${F}_{cor}$, on two representative slices is discussed here (figure 6c–f, iii and iv). The ${F}_{pre}$ and ${F}_{cor}$ are examined here as possible driving forces of spanwise flow, whereas the contribution of centrifugal acceleration is ignored. This is because the centrifugal acceleration is weaker than the pressure gradient and Coriolis acceleration and remains constant within the cylindrical slice at 0.25b or 0.5b. In addition, since the strong spanwise flow is mostly located above the upper surface of the wing (figure 6c–f, A-ii), the ${F}_{pre}$ and ${F}_{cor}$, instead of the centrifugal acceleration, are more likely to be responsible for the formation of spanwise flow. It can be observed that a strong spanwise flow exists within the LEV at 0.25b (figure 6c–f, A-ii) and extends into the wake and covers the entire dorsal surface of the wing at 0.5b (figure 6c–f, B-ii).

Moreover, an apparent tip-ward pressure gradient (${F}_{pre}$) exists near the downstream edge of the LEV at 0.25b (figure 6c–f, A-iii), overlapped with regions of the local spanwise flow. However, the ${F}_{pre}$ inside the LEV is reversed at 0.5b and the local tip-ward ${F}_{pre}$ region becomes dispersed behind the LEV (figure 6c–f, B-iii). The contribution of ${F}_{cor}$ in the formation of spanwise flow is significantly weaker than that of ${F}_{pre}$, although the tip-ward ${F}_{cor}$ region shows a good agreement with the local tip-ward spanwise flow at both 0.25b and 0.5b (figure 6c–f, A-iv and B-iv). Comparing the regions of $C_r^{(+)}$ and ${F}_{pre}$, it is remarkable that $C_r^{(+)}$ regions can be well predicted by those of tip-ward ${F}_{pre}$ at both 0.25b and 0.5b, which suggests that the reduction of $C_r^{(+)}$ around the mid-span (beyond which LEV becomes less stable) can be attributed to the lack of tip-ward pressure gradient. Therefore, despite the fact that the tip-ward spanwise flow is usually considered as a source of tip-ward vorticity convection (Ellington et al. Reference Ellington, Van Den Berg, Willmott and Thomas1996; Wu & Sun Reference Wu and Sun2004), our result suggests that the tip-ward vorticity convection is strongly related to the establishment of tip-ward pressure gradient.

Apart from convections of radial vorticity, the vortex tilting and stretching also contribute to the LEV stability. We examined the vortex tilting and stretching at two representative slices at $\textit {Re}=1500$ (figure 7, slices A and B), including a decomposition into four components (panels i to iv showing $T_{t\rightarrow r}$, $T_{y \rightarrow r}$, PVT$_{r}$ and S$_r$, respectively). The integrals of vortex tilting and stretching within the LEV region are plotted over the span (i.e. $\hat {T}_{t\rightarrow r}$, $\widehat {\mathrm {PVT}}_{r}$ and $\hat {S}_{r}$). The overall contribution of vortex tilting and stretching can balance the LEV vorticity around the mid-span (first column in figure 7b–d). When separating into individual components, the $T_{t\rightarrow r}$ and PVT$_{r}$ are dominated by their positive subterms (T$_{t\rightarrow r}^{(+)}$ and PVT$_{r}^{(+)}$) in the LEV region and consistently remove LEV vorticity within the mid-span (second and fourth columns in figure 7b–d).

Figure 7.

Vortex tilting and stretching in the radial direction at $\textit {Re}=1500$: (a) $\lambda =1.5$, (b) $\lambda =4$, (c) $\lambda =5$ and (d) $\lambda =6$. The representative slices A and B are defined in figure 6. Here, $\hat {T}_{t\rightarrow r}$, $\widehat {\mathrm {PVT}}_{r}$ and $\hat {S}_{r}$ are the integrals of $T_{t \rightarrow r}$, PVT$_{r}$ and $S_r$ within the LEV region.

Among all vortex tilting and stretching in the radial direction, $T_{t\rightarrow r}$ has the strongest intensity and occurs within the entire half-span from the root ($\hat {T}_{t \rightarrow r}$ in figure 7a–d). The contribution of PVT$_{r}$ remains consistent along the entire span but at a considerably lower magnitude ($\widehat {\mathrm {PVT}}_{r}$ in figure 7a–d). Moreover, PVT$_{r}$ exhibits low spatial–temporal variations, indicating that it is a consistent mechanism removing LEV vorticity during both the formation period and steady state. In contrast, the evolution of S$_r$ is more complex. The $\hat {S}_{r}$ at 0.25b indicates a vortex stretching (enhancement) of radial vorticity during the LEV formation (A-iv in figure 7a–d) whereas a strong vortex compression (removal) of radial vorticity is observed around the mid-span during $\lambda =4 - 6$ (B-iv in figure 7b–d). From the $\hat {S}_{r}$ in figure 7, the peak of vortex compression is generated around the wing tip at $\lambda =1.5$ and then moves towards the mid-span until the steady state. This inward movement of $\hat {S}_{r}$ peak can compensate for the local reduction of tip-ward vorticity convection (figure 6d–f), and thus helps to remove the local LEV vorticity.

From above, $T_{t\rightarrow r}$ is one of the most important contributors to LEV stability at $\textit {Re}=1500$. On the other hand, the vortex tilting from radial vorticity into the tangential direction ($T_{r\rightarrow t}$), as indicated by the 3-D LEV structure in figure 5(a), contributes to the development of tangential vorticity. Thus, the tangential vorticity transport terms within the LEV are examined. Here, the convection of tangential vorticity is omitted and only the tangential components of vortex tilting in (2.7), including $T_{r\rightarrow t}$, $T_{y\rightarrow t}$ and PVT$_{t}$, are discussed. Since the tangential vorticity within the LEV is mostly positive (figure 8a), positive values of $T_{r\rightarrow t}$, $T_{y\rightarrow t}$ and PVT$_{t}$, i.e. $T_{r\rightarrow t}^{(+)}$, $T_{y\rightarrow t}^{(+)}$ and PVT$_{t}^{(+)}$, indicate an increase of local tangential vorticity. As shown in figure 8(a), $T_{r\rightarrow t}^{(+)}$ and PVT$_{t}^{(+)}$ tend to enhance the tangential vorticity in the inboard region. In the outboard region, PVT$_{t}^{(+)}$ is the main source that enhances local tangential vorticity. According to the derivation of Werner et al. (Reference Werner, Chung, Wang, Liu, Cimbala, Dong and Cheng2019), the PVT$_{t}$ is mathematically related to the vertical gradient of radial Coriolis acceleration. Thus, the radial Coriolis acceleration is found to be strongly related to the increase of tangential vorticity.

Figure 8.

The balance of radial and tangential vorticity at $\textit {Re}=1500$. (a) Vortex tilting in the evolution of tangential vorticity and (b) a schematic of primary stabilizing mechanisms (SMs) in the stable LEV region. The isosurfaces are drawn by $\omega _t^{*}=2$ and ${Q}=2$.

According to the $T_{t\rightarrow r}^{(+)}$ and $T_{r\rightarrow t}^{(+)}$ shown in figures 7 and 8, there exists a conversion between radial vorticity and tangential vorticity within the mid-span, resembling a negative feedback loop that stabilizes the vorticity in both directions. When the local radial vorticity (tip-ward) accumulates, the local tangential vorticity also increases via induced vortex tilting ($T_{r\rightarrow t}^{(+)}$). This further enhances the vortex tilting of tangential vorticity into the positive radial direction ($T_{t\rightarrow r}^{(+)}$), which is opposite to the radial vorticity in the LEV. This conversion functions as a negative feedback loop that attenuates the increase of local tip-ward radial vorticity, leading to a dynamic equilibrium, i.e. a steady state. Therefore, it is likely that the stable LEV in the inboard region can be mainly attributed to this mechanism, which we name as ‘radial–tangential vorticity balance’ (RTVB). In summary, the stable LEV in the inboard region at $\textit {Re} =1500$ is mainly led by three stabilizing mechanisms (SMs), including tip-ward vorticity convection, RTVB and planetary vorticity tilting (figure 8b). In terms of the intensity of these SMs, the RTVB contributes most to the inboard LEV stability (see $T_{t\rightarrow r}^{(+)}$ in figure 7). It should be noted that the tangential tilting of planetary vorticity, which is related to the radial Coriolis acceleration, is one of the main sources of tangential vorticity, and therefore also contributes towards the LEV stability, albeit indirectly, via the RTVB mechanism. The RTVB and the PVT$_{r}$ are sustained within the entire stable LEV region whereas the tip-ward vorticity convection becomes diminished around the mid-span at the steady state (figure 6). The local reduction of tip-ward vorticity convection around mid-span is compensated by the increase of vortex compression of radial vorticity (figure 7).

3.2.2. Sectional analysis at $\textit {Re}=100$

As $\textit {Re}$ decreases to 100, the stable LEV region is extended towards the wing tip while the conical LEV structure becomes less compact (figure 5a). Meanwhile, the strength of most vorticity transport terms (except PVT$_{r}$) is reduced. As shown in figure 9, the tangential convection and the vertical convection ($C_{t}$ and $C_{y}$) resemble their counterparts at $\textit {Re}=1500$, whereas the 3-D structures become more stable (figure 9a,b). Moreover, the tip-ward vorticity convection is also present (outlined by the positive $C_{r}$, i.e. $C_{r}^{(+)}$, in figure 9c–e) and therefore contributes to LEV stability at $\textit {Re}=100$. However, while the $C_{r}^{(+)}$ region extends beyond the mid-span during LEV formation, it becomes more confined towards 0.25b at the steady state (figure 9e). Moreover, the spanwise flow (tip-ward) covers most of the dorsal wing surface at $\textit {Re}=100$, which is not well overlapped with the LEV region (figure 9c–e, A-ii and B-ii). Note that an apparent tip-ward spanwise flow is formed at the mid-span whereas less local tip-ward vorticity convection is observed (figure 9c–e). Similar to the $\textit {Re}=1500$ case, the $C_{r}^{(+)}$ region can also be mostly predicted by that of tip-ward pressure gradient (${F}_{pre}$) at $\textit {Re}=100$. For the generation of spanwise flow, the contributions of ${F}_{pre}$ and radial Coriolis acceleration (${F}_{cor}$) are comparable. Since the tip-ward ${F}_{cor}$ region shows a good agreement with tip-ward spanwise flow, it can be inferred that the spanwise flow at $\textit {Re}=100$ is mainly driven by radial Coriolis acceleration, the mechanism of which is akin to the Ekman pump as described by Lentink & Dickinson (Reference Lentink and Dickinson2009).

Figure 9.

Convection of radial vorticity at $\textit {Re}=100$: (a) steady-state $C_t$, (b) steady-state $C_y$ and (c–e) evolution of $C_r$. Details about the various panels can be found in figure 6.

The evolution of vortex tilting and stretching is further examined at $\textit {Re}=100$ (figure 10). Note that the results at $\lambda =0.5$ are insignificant and are not shown here. As $\textit {Re}$ decreases to 100, $T_{t\rightarrow r}^{(+)}$ and PVT$_{r}^{(+)}$ become the two dominant vorticity transport mechanisms that contribute to LEV stability (figure 10b,c). Unlike the $\textit {Re}=1500$ case, the intensity of PVT$_{r}^{(+)}$ is now stronger than that of $T_{t\rightarrow r}^{(+)}$ within the LEV (shown by $\hat {T}_{t\rightarrow r}$ and $\widehat {\mathrm {PVT}}_{r}$). The LEV vorticity at 0.25b is mainly removed by PVT$_{r}^{(+)}$ while both $T_{t\rightarrow r}^{(+)}$ and PVT$_{r}^{(+)}$ contribute to LEV stability at 0.5b. When comparing $\widehat {\mathrm {PVT}}_{r}$ at $\textit {Re}=100$ and 1500 (figures 7 and 10), it is found that the intensity and spanwise distribution of $\widehat {\mathrm {PVT}}_{r}$ remain almost identical as $\textit {Re}$ decreases, indicating that the contribution of PVT$_{r}$ in LEV stability is almost $\textit {Re}$-independent.

Figure 10.

Vortex tilting and stretching in the radial direction at $\textit {Re}=100$: (a) $\lambda =1$, (b) $\lambda =2$ and (c) $\lambda =4$. The representative slices A and B are defined in figure 9.

To further examine the $\textit {Re}$ dependence of the RTVB mechanism, the temporal variation of tangential vorticity and the contribution of vortex tilting at $\textit {Re}=100$ are shown in figure 11. Since the vortex stretching becomes negligible at $\textit {Re}=100$ (figure 10), no discussion on vortex stretching is presented here. It is shown that the vortex tilting from radial vorticity into the tangential direction ($T_{r\rightarrow t}^{(+)}$) exists in the entire LEV region during its formation ($\lambda <2$) and stabilizes within the inboard region at the steady state. Together with the strong $T_{t\rightarrow r}^{(+)}$ shown in figure 10, it can be seen that the RTVB mechanism also contributes to the LEV stability at $\textit {Re}=100$. Moreover, the PVT$_{t}$, which is related to the radial Coriolis acceleration, contributes consistently towards the development of tangential vorticity at both $\textit {Re}=100$ and 1500. In summary, as $\textit {Re}$ decreases to 100, the three SMs illustrated in figure 8(b) still all contribute to the inboard LEV stability, but with varied relative magnitudes. The tip-ward vorticity convection (SM1) and the RTVB (SM2) are significantly reduced in strength at $\textit {Re}=100$, while the intensity and distribution of the radial tilting of planetary vorticity (SM3) are almost $\textit {Re}$-independent between $\textit {Re}=100$ and 1500.

Figure 11.

Vortex tilting in the evolution of tangential vorticity at $\textit {Re}=100$. The isosurface is drawn by $\omega _t^{*}=1$.

3.2.4. Mechanisms contributing to LEV stability: a spatial–temporal analysis

The spatial–temporal variation of primary vorticity convection and vortex tilting and stretching terms at varying $\textit {Re}$ is summarized in figure 13. Five representative slices are presented to demonstrate the spatial variation along the span, and the integrals of positive subterms in $C_r$, $T_{t\rightarrow r}$, PVT$_{r}$ and $S_r$, i.e. $\hat {C}_r^{+}$, $\hat {T}_{t\rightarrow r}^{+}$, $\widehat {\mathrm {PVT}}_{r}^{+}$ and $\hat {S}_r^{+}$, are considered here. First, it is evident that the radial tilting of planetary vorticity ($\widehat {\mathrm {PVT}}_{r}^{+}$) not only is almost $\textit {Re}$-independent (when $\textit {Re}$ is higher than 100) but also has the least spatial (spanwise) and temporal variations when fully developed (figure 13c). In contrast, tip-ward vorticity convection ($\hat {C}_{r}^{+}$) is mostly confined within the mid-span and becomes stronger when $\textit {Re}>1000$ (figure 13a). Note that the strong $\hat {C}_{r}^{+}$ at 0.9b is attributed to the interaction with the tip vortex. At $\textit {Re}>1000$, $\hat {C}_{r}^{+}$ beyond the mid-span (at the steady state) is absent while vortex compression ($\hat {S}_{r}^{+}$ in figure 13d) assists in stabilizing the local LEV. Once the $\textit {Re}$ is below 100, the $\hat {C}_{r}^{+}$ and $\hat {S}_{r}^{+}$ are both negligible.

Figure 13.

The effects of Re on the spatial–temporal evolution of vorticity convection, vortex tilting and stretching that contribute to LEV stability: (a) $\hat {C}_{r}^{+}$, (b) $\hat {T}_{t\rightarrow r}^{+}$, (c) $\widehat {\mathrm {PVT}}_{r}^{+}$ and (d) $\hat {S}_{r}^{+}$.

The $\textit {Re}$ effect on the radial tilting of tangential vorticity (shown by $\hat {T}_{t\rightarrow r}^{+}$) is strong and there is a steep reduction of intensity as $\textit {Re}$ decreases from 500 to 100 (figure 13b). The main $\hat {T}_{t\rightarrow r}^{+}$ region is mostly located within the mid-span at $\textit {Re}>1000$. The RTVB illustrated in figure 8 is related to this strong $\hat {T}_{t\rightarrow r}^{+}$. Consequently, the inboard LEV stability of revolving wings at $\textit {Re}>1000$ can be mainly attributed to the tip-ward vorticity convection, RTVB and the radial tilting of planetary vorticity, despite the fact that the vortex compression also contributes to LEV stability near mid-span. For $\textit {Re}$ around 100, the radial tilting of planetary vorticity and RTVB become the two main mechanisms.

As $\textit {Re}$ further decreases to 10, the effects of vorticity convection and vortex tilting and stretching in LEV formation and stability become negligible (which can be inferred from figure 13), indicating that the vorticity generated from the leading edge remains close to the rim. Therefore, the 3-D LEV structure at $\textit {Re}=10$ is closely aligned with the leading edge and no conversion of radial vorticity and tangential vorticity is observed (figure 5a). In our following discussion, it is found that viscous diffusion (D in (2.9)) regulates the increase of radial vorticity within the LEV at $\textit {Re}=10$. The positive viscous diffusion within the LEV region is integrated ($\hat {D}$ in figure 14a) and a linear increase of $\hat {D}$ over the span is observed at both $\textit {Re}=10$ and 100. The viscous diffusion in the LEV region is reduced as $\textit {Re}$ increases and becomes almost negligible at $\textit {Re}>1000$. According to a spatial–temporal analysis of integrated viscous diffusion within the LEV, i.e. $\tilde {D}$ (figure 14b), the contribution of viscous diffusion in LEV stability is comparable to the PVT$_{r}$ and RTVB when $\textit {Re}$ is about 200 (the legends of figures 13 and 14 are identical). For revolving wings with $\textit {Re}<200$, the viscous diffusion starts to dominate the evolution of radial vorticity. Since other vorticity transport mechanisms are almost negligible at $\textit {Re}=10$, the conical LEV structure at this $\textit {Re}$ level is maintained under the balance of vorticity shedding from the leading edge and viscous diffusion. Note that the $\tilde {D}$ can experience a transient overshoot at the end of acceleration ($\lambda =0.5$) at $\textit {Re}<100$, as shown by the star in figure 14(b).

Figure 14.

The effects of $\textit {Re}$ on viscous diffusion within the LEV region: (a) spanwise distribution and (b) spatial–temporal analysis. The plain line in figure 13(b) shows the contour line at $\tilde {D}=2.5$.