3.1. Tip effects

Tip effects play an important role in the development of wakes behind low-aspect-ratio wings. At $sAR=0.5$, steady flows are observed up to $\alpha =26^{\circ }$, as shown in figure 3(a). These steady flows typically feature a pair of tip vortices, which induce strong downwash over the wing span, suppressing the formation of large LEVs. Such a mechanism is responsible for the stability of the wake, particularly for low-aspect-ratio unswept wings (Taira & Colonius Reference Taira and Colonius2009; DeVoria & Mohseni Reference DeVoria and Mohseni2017; Zhang et al. Reference Zhang, Hayostek, Amitay, He, Theofilis and Taira2020). At higher sweep angles, the vortical structures that emanate from the leading edge grow in size, covering a significant portion of the suction surface of the wing. These vortical structures are also beneficial to the stability of the wake, as will be discussed in detail in § 3.2. Unsteady flows are observed only at $\alpha \gtrsim 30^{\circ }$ for $\varLambda =0^{\circ }\text {--}37.5^{\circ }$. These unsteady wakes are characterized by the periodic shedding of hairpin vortices (Taira & Colonius Reference Taira and Colonius2009).

As the aspect ratio is increased, the downwash induced by the tip vortices weakens along the midspan, allowing the roll-up of the vortex sheet at the leading edge. As a result, unsteady vortex shedding develops near the midspan, as visualized in figure 2(b). The stability boundary (dashed line in figure 3) shifts towards lower angles of attack at $sAR=1$ and 2 for low-swept wings. Additional details on the three-dimensional unsteady wake dynamics of wings under tip effects are reported in Taira & Colonius (Reference Taira and Colonius2009) and Zhang et al. (Reference Zhang, Hayostek, Amitay, He, Theofilis and Taira2020).

3.2. Midspan effects

For wings with larger aspect ratio ($sAR\gtrsim 1$), in place of the weakened tip effects, the midspan symmetry introduces another type of three-dimensionality that dominates the wake. We present the skin-friction lines for cases of $(\alpha ,sAR)=(20^{\circ },1)$ with varying $\varLambda$ as shown in figure 4. With the increase in sweep angle, the leading-edge boundary layer separation point near the midspan gradually shifts towards the trailing edge. This is caused by the growth of the vortical structures emanating from the leading edge, as observed in figure 3(b). The increase in sweep angle for $sAR\gtrsim 1$ also leads to the attenuation of the tip vortices, as reflected by the diminishing three-dimensional skin-friction line pattern near the wing tip in figure 4. In addition, the reattachment of the separated recirculating flow, which occurs near the trailing edge, shifts away from the midspan towards the tip with increasing sweep angle. The above observations suggest a switch-over of the source of three-dimensionality from the wing tip in low-$\varLambda$ cases to the midspan in high-$\varLambda$ cases.

Figure 4.

Skin-friction lines on the suction side of wings with $(\alpha ,sAR)=(20^{\circ },1)$.

Let us take a closer look at the vortical structures near the midspan. A representative case of $(\alpha ,sAR,\varLambda )=(20^{\circ }, 2, 45^{\circ })$ is shown in figure 5. For this case, the vortex sheet rolls up along the spanwise direction, covering the entire chord over the inboard section of the wing. Owing to the symmetry condition, the identical vortical structures on the two sides of the mid-plane are oriented at an angle of $180^{\circ } - 2\varLambda$, which in the current case is $90^{\circ }$. As a result, each of the vortical structures is subjected to the downward velocity (pointing at $-y$ direction) induced by its symmetric peer on the other side of the midspan. This is clearly manifested in figure 5(b) by the strong negative cross-flow velocity $u_y$ on the suction side near the $z=0$ plane.

Figure 5.

Wake for the case $(sAR, \alpha , \varLambda )=(2, 20^{\circ }, 45^{\circ })$. (a) Vortical structures visualized by isosurface of $Qc^2/U_{\infty }^2=1$. The thick solid lines represent the approximate location of vortical structures near the midspan, with the curved arrows showing their directions of rotation. The dashed lines indicate the locations of visualizations shown on the right. (b,c) The $u_y$ and $u_z$ fields, respectively, at different $z$ locations.

Owing to the three-dimensional midspan effects described above, the vortical structures near the midspan are stabilized by the downward velocity induced from themselves. For cases with smaller $\varLambda$, the angle between the symmetric vortical structure tends towards $180^{\circ }$, and the mutually induced velocity becomes smaller. The three-dimensionality developed from the midspan is able to stabilize the wakes of a considerable number of cases, as labelled by the green triangle in figure 3. The midspan effects are also observed for swept wings with $sAR=0.5$, in which cases the tip vortices also stabilize the wake. In fact, the strengthening of the tip vortices with increasing $\varLambda$ at $sAR=0.5$ (see figure 3a, $\alpha =26^{\circ }$, for example) is likely to be related to the three-dimensional midspan effects. As vortex sheets emanating from the leading edge are less likely to roll up due to the midspan effects, they counteract less on the formation of tip vortices. We note that spanwise vorticity transport, which is considered as the key mechanism for the formation of steady LEVs on delta wings (Polhamus Reference Polhamus1966; Gursul Reference Gursul2005), does not play an important role in the formation of the midspan vortical structures in the current case. This is manifested by the fact that the strong outboard velocity $u_z$ does not coincide with the vortex core, as shown in figure 5(c).

The downward induced velocity described above is strong near the midspan, as shown in figure 5(b). With the gradual weakening of the midspan effects towards the outboard sections, unsteadiness develops locally near the tip region, while the midspan region still remains steady. The resulting flows resemble the ‘tip stall’ phenomenon, as described in Black (Reference Black1956), Zhang et al. (Reference Zhang, Jaworski, McParlin and Turner2019) and Visbal & Garmann (Reference Visbal and Garmann2019). This type of flow, as indicated by yellow squares in figure 3, prevails for swept wings with large aspect ratios (e.g. $sAR=4$) and high angles of attack (e.g. $\alpha =26^{\circ }\text {--}30^{\circ }$). For wings with large $sAR$, the unsteady shedding develops from the leading edge and features mostly spanwise vortices (see figure 2e, for example). In contrast, for wings with low aspect ratio but large angles of attack, the unsteady vortices are shed from the wing tip, and they feature hairpin structures with the legs originated from the pressure and suction sides of the wing, as shown in figure 2(f).

3.3. Formation of streamwise finger-like structures

For high sweep angles of $\varLambda \gtrsim 37.5^{\circ }$, the flows over high-aspect-ratio wings ($sAR \gtrsim 2$) transition from the unsteady tip shedding to steady wakes, through the formation of the streamwise finger-like structures. As shown in figure 2(f), the finger-like structures are oriented at an angle higher than $\varLambda$ with respect to the $z$ axis. These structures bend towards the streamwise direction away from the wing. Wakes of swept wings with even $sAR$ as large as 10 are stabilized with the alternating formation of the streamwise structures along the wing span, as shown in figure 6(a).

Figure 6.

Streamwise vortices along high-aspect-ratio wings with $\varLambda =45^{\circ }$. (a) Vortical structures visualized by isosurface of $Qc^2/U_{\infty }^2=1$ for $sAR=4$, 6 and 10. (b) The $u_x$ field at indicated slices as shown in (a). The red lines represent contours of $Qc^2/U_{\infty }^2=0.5$ with the curved arrows indicating the direction of rotation. (c) The $u_z$ field on the slices IV for cases with three aspect ratios. Black circles show the isosurfaces of $Qc^2/U_{\infty }^2=0.5$.

These streamwise vortices observed in the present work are formed under the same mechanism as those in the wakes of axisymmetric slender bodies at high incidence (Sarpkaya Reference Sarpkaya1966; Thomson & Morrison Reference Thomson and Morrison1971). The impulse flow analogy has long been used to understand these vortical structures. According to this analogy, the progressive development of the wake along the wing span when viewed in cross-flow planes is similar to the temporal growth of the flow behind a 2-D wing translated impulsively from rest. Three slices at different spanwise locations in figure 6(b) show a temporal-like evolution of wake vortices along the wing span. In addition, the spacing between the neighbouring streamwise vortices is fixed at $g = 2.4$, which follows $g \approx U_{\infty } \sin (\varLambda ) T_{2D}$, in which $U_{\infty }\sin (\varLambda )$ represents the speed of the spanwise flow, and $T_{2D}=3.2$ is the non-dimensional period of vortex shedding in the analogous 2-D case (for an infinite swept wing). This relationship suggests a close analogy between the 2-D unsteady flow and the present three-dimensional steady flow, where the time dependence of the former is replaced by the spatial dependence of the latter.

We also plot the velocity component $u_z$ on the planes that are parallel to the wing span in figure 6(c). The profiles of $u_z$, as well as the arrangement of the streamwise vortices, are almost identical along the spanwise direction for the wakes behind the wings with the three aspect ratios. These finger-like vortices are positioned at two sides of strong outboard velocity $u_z$, resembling the configuration where an inverse von Kármán vortex street is formed in a jet profile. This observation suggests the important role of the spanwise velocity in the formation of steady three-dimensional vortices. Similar streamwise structures have also been reported for rotating large-$AR$ wings (Jardin Reference Jardin2017), in which spanwise flow is promoted by artificially adding Coriolis force.

3.4. Aerodynamic forces

The vortical structures described above influence the aerodynamic forces on the swept wings. We examine the effects of sweep angle on the distribution of the time-averaged sectional lift coefficients $\overline {C_l}$ for the representative cases of $(\alpha ,sAR)=(20^{\circ },2)$ in figure 7(a). For the unswept wing, the sectional lift increases slowly from the midspan towards outboard, reaching a maximum at $z\approx 1.5$ before its drastic decrease at the tip. The slight swell-up of the sectional lift at the outboard location is attributed to downwash from the tip vortices (DeVoria & Mohseni Reference DeVoria and Mohseni2017; Zhang et al. Reference Zhang, Hayostek, Amitay, He, Theofilis and Taira2020). For swept wings, the sectional lift coefficients are significantly higher at the inboard locations. Such high lift is achieved due to the midspan effects, which trap additional vorticity in the vicinity of the suction surface (as described in § 3.2), thus increasing the local circulation. We note that the lift distribution of swept wings in high-$Re$ flows exhibits the opposite pattern, where sectional lift is low near midspan and high at outboard sections (Schlichting & Truckenbrodt Reference Schlichting and Truckenbrodt1960; Nickel & Wohlfahrt Reference Nickel and Wohlfahrt1994). Moreover, for the same angle of attack, one sees lift loss everywhere along the span of the swept wing compared with the unswept wing (Visbal & Garmann Reference Visbal and Garmann2019). These differences highlight the midspan effects in augmenting lift of swept wings in low-Reynolds-number separated flows.

Figure 7.

Force coefficients for $\alpha =20^{\circ }$. (a) Sectional lift coefficients for varied sweep angles at a fixed aspect ratio $sAR=2$. (b) Sectional lift coefficients for cases of $\varLambda =45^{\circ }$ with different aspect ratios. (c,d) Lift coefficients $\overline {C_L}$ and drag coefficients $\overline {C_D}$, respectively, over sweep angle $\varLambda$ for different aspect ratios.

The effects of aspect ratio on the distribution of sectional lift coefficients of swept wings are examined for the cases $(\alpha ,\varLambda )=(20^{\circ },45^{\circ })$ in figure 7(b). A monotonic decrease of sectional lift is observed across $sAR=0.5$ to 6. The maximum sectional lift at midspan $z=0$ increases with the aspect ratio, and eventually saturates as $sAR$ reaches 3. This observation suggests that the tip effect is still noticeable for low-$AR$ swept wings even though distinct tip vortices are not observed. For high-$AR$ wings, the $C_l$–$z$ curves feature steep slope over the inboard region, reflecting the weakening of the midspan effects. The slope becomes shallower towards the outboard, where midspan effects diminish. The streamwise finger-like structures described in § 3.3 do not have a noticeable influence on the lift, as indicated by the flatness of the $C_l$–$z$ curves at the outboard span shown in figure 7(b).

Finally, we present the total lift $\overline {C_L}$ and drag $\overline {C_D}$ coefficients of the swept wings at $\alpha =20^{\circ }$ with different aspect ratios and varying sweep angles in figure 7(c,d). For swept wings with $sAR=0.5$ and 1, the lift coefficients increase with sweep angle, due to the dominance of the midspan effects over the entire wing span. The drag coefficients of the low-aspect-ratio wings, on the other hand, do not show significant variations over the increase of sweep angle. It is interesting to note that $\overline {C_D}$ for $sAR=0.5$ is higher than that for $sAR=1$, an observation also reported in Taira & Colonius (Reference Taira and Colonius2009). For wings with $sAR\gtrsim 3$, the contribution of the high sectional lift at the inboard region to the total lift force is overshadowed by the lower sectional lift at the outboard region. As a result, the lift coefficients of wings with high aspect ratio decrease with increasing sweep angle. A similar trend is also observed for the drag coefficients of high-aspect-ratio wings. While for wings with small $\varLambda$ the lift coefficients $\overline {C_L}$ are positively related with $sAR$, at the highest sweep angle of $\varLambda =45^{\circ }$, the maximum lift coefficient is achieved with $sAR=2$, and is even higher than the analogous 2-D case. The midspan effects as lift enhancement mechanism for low-aspect-ratio swept wings could potentially inspire designs of high-lift devices.