3.1 Boundary vorticity flux and skin-friction lines

We begin our analysis by examining the introduction of vorticity to the flow from the wing surface. In incompressible flow, vorticity is only generated on the solid surface and diffuses into the flow (Morton Reference Morton1984; Hornung Reference Hornung1989; Wu, Ma & Zhou Reference Wu, Ma and Zhou2007). The rate at which vorticity is created at the surface is given by the wall-normal boundary vorticity flux

(3.1)$$\begin{eqnarray}\unicode[STIX]{x1D748}=-\unicode[STIX]{x1D708}\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D72E},\end{eqnarray}$$

where $\unicode[STIX]{x1D72E}=(\unicode[STIX]{x1D735}\unicode[STIX]{x1D74E})_{0}$ is the vorticity gradient tensor at the surface.

Figure 6.

Time-averaged flow behaviour at the wing surface for $(\unicode[STIX]{x1D6FC},sAR)=(22^{\circ },4)$. (a) Three components of the wall-normal boundary vorticity flux. The surface of the wing is unfolded to a plane by cutting along the trailing edge, TE. The dashed line indicates the leading edge, LE, of the wing. The insets show the boundary vorticity flux on the side surface. (b) Skin-friction lines on the suction surface.

Let us show the vorticity flux along the wing surface for a representative case of $(\unicode[STIX]{x1D6FC},sAR)=(22^{\circ },4)$. The time-averaged wall-normal boundary vorticity flux components $\overline{\unicode[STIX]{x1D70E}}_{x}$, $\overline{\unicode[STIX]{x1D70E}}_{y}$ and $\overline{\unicode[STIX]{x1D70E}}_{z}$ are plotted in figure 6(a) to identify the local generation of vorticity. As expected, the spanwise vorticity flux $\overline{\unicode[STIX]{x1D70E}}_{z}$ dominates over the surface of the wing. Negative $\overline{\unicode[STIX]{x1D70E}}_{z}$ is contained within a narrow region along the leading edge of the wing. This region injects a large amount of negative $\unicode[STIX]{x1D714}_{z}$ into the flow, forming the leading-edge vortex sheet. Compared to $\overline{\unicode[STIX]{x1D70E}}_{z}$, the other two components of the vorticity flux appear negligible for the majority of the span. It is only at the close vicinity of the wing tip that considerable level of $\overline{\unicode[STIX]{x1D70E}}_{x}$ and $\overline{\unicode[STIX]{x1D70E}}_{y}$ can be found. The creation of vorticity on the finite-aspect-ratio wing is not strongly affected by the tip effects and is predominantly two-dimensional along the span. There is also vorticity generation on the side surface of the wing tip. However, vorticity generated from the side does not appear to play a major role. Similar tip vortical structures are also found in simulated flows over finite flat-plates of zero thickness (Taira & Colonius Reference Taira and Colonius2009).

Once the vorticity is generated at the wing surface, it is diffused into the flow and is convected by the free stream. The developed flow imprints the skin-friction field $\unicode[STIX]{x1D749}$ on the wall, as shown in figure 6(b). Three regions can be recognized from the skin-friction field. The region close to the leading edge away from the tip is occupied by the nearly two-dimensional boundary layer, which separates from the wing in a uniform manner. Post separation, a region of reversed flow appears on the suction side of the wing where the skin-friction lines exhibit spanwise component of the flow. The dominant three dimensionality is observed over the wing tip region with a three-dimensional swirl pattern, as reported in the experimental work of Winkelmann & Barlow (Reference Winkelmann and Barlow1980).

Figure 7.

Time-averaged skin-friction line patterns on the suction side for the cases of $sAR=4$ for (a) $\unicode[STIX]{x1D6FC}=8^{\circ }$, (b) $\unicode[STIX]{x1D6FC}=14^{\circ }$, (c) $\unicode[STIX]{x1D6FC}=20^{\circ }$ and (d) $\unicode[STIX]{x1D6FC}=26^{\circ }$.

To assess the influence of the angle of attack on the near-surface flow pattern, we visualize the time-averaged skin-friction lines for the cases of $sAR=4$ with different $\unicode[STIX]{x1D6FC}$ in figure 7. For higher angles of attack, the flow separation occurs more upstream, and the spanwise extent of the boundary layer that is affected by the tip effects becomes smaller. The skin-friction lines in the recirculation region becomes more three-dimensional at higher $\unicode[STIX]{x1D6FC}$.

The development of the three-dimensional flow can be envisioned from the boundary vorticity flux and skin-friction lines. Vorticity is introduced from the wing surface in a predominantly two-dimensional manner. The vortex sheets emanate from the leading edge and the wing tip. While the roll-up of the leading-edge vortex sheet is two-dimensional, the roll-up from the tip is fundamentally three-dimensional yielding a streamwise tip vortex. As the tip vortex develops, its influence grows from being locally confined to imposing spanwise variations, as evident in figure 5. We discuss the three-dimensional wake dynamics in the following sections.

Figure 8.

Top view of the wake for the case of $(\unicode[STIX]{x1D6FC},sAR)=(22^{\circ },4)$. The black lines represent the vorticity lines. Isosurfaces of $Q=1$ are shown in transparent grey.

3.2 Three-dimensional wake structures

3.2.1 An overview of the vortical structures

Let us continue to examine the three-dimensional wake structures for the representative case of $(\unicode[STIX]{x1D6FC},sAR)=(22^{\circ },4)$. A top view of the wake vortical structures is shown in figure 8. As also seen in figure 5, the wake can be divided into the tip vortex region at the free end and the unsteady vortex shedding region near the midspan. The tip vortex appears practically steady over time in the current case. The unsteadiness in the tip vortex becomes noticeable for cases with smaller $sAR$ and larger $\unicode[STIX]{x1D6FC}$, as discussed later in §§ 3.2.2 and 3.2.3. The unsteady region of the wake can be further divided into two subregions. Adjacent to the midspan ($z\lesssim 1.5$), the wake is dominated by vortex cores that are aligned predominantly with the spanwise direction. These vortices are formed by the roll-up of the vortex sheets from the leading and trailing edges, and shed alternatingly into the wake. On the other hand, the shedding vortices at $2\lesssim z\lesssim 2.5$ feature braid-like structures. These vortical structures are comprised of streamwise vorticity $\unicode[STIX]{x1D714}_{x}$ and cross-flow vorticity $\unicode[STIX]{x1D714}_{y}$, and are aligned mostly perpendicular to the spanwise direction. These vortical structures shed at the same frequency with the spanwise vortices.

The complex vortical structures in the unsteady region can be better understood by visualizing the vorticity lines, as depicted by the black solid lines in figure 8. Each vortex core in the braid-like region connects with a pair of counter-rotating spanwise vortical structures. These structures together form a closed vortex loop (considering the symmetry boundary condition at the midspan). The unsteady wake region is constituted of a series of such vortex loops arranged in a zigzag manner. It is thus clear that the braid-like structures play the role of closing the spanwise vortex system. This is in accordance with Helmholtz’s second law (Helmholtz Reference Helmholtz1858) which states that a vortex filament cannot terminate in a fluid (Batchelor Reference Batchelor1967). Similar vortex line models have also been proposed to describe the wakes of finite-span circular cylinders (Taneda Reference Taneda1952; Levold Reference Levold2012).

Figure 9.

Perspective views of the wake vortices behind the finite wings with $sAR=1$, 2, 4 and 6, $\unicode[STIX]{x1D6FC}=14^{\circ }$, $22^{\circ }$ and $30^{\circ }$. Vortical structures are visualized by the isosurfaces of $\Vert \unicode[STIX]{x1D74E}\Vert =2$ in light grey and $Q=1$ in dark grey.

3.2.2 Effects of $sAR$ and $\unicode[STIX]{x1D6FC}$ on the wake

Separated flows behind finite-aspect-ratio wings are strongly influenced by the aspect ratio $sAR$ and the angle of attack $\unicode[STIX]{x1D6FC}$. Let us show some representative wakes for a range of aspect ratios and angles of attack in figure 9. The short wings with $sAR=1$ generate different wake features compared with those discussed in § 3.2.1. Stable three-dimensional flow is observed at $\unicode[STIX]{x1D6FC}=14^{\circ }$, while the analogous two-dimensional flow at the same angle of attack exhibits periodic unsteady shedding. This indicates again that the tip effects stabilize the local flow, which in the current case of $sAR=1$ is the entire span. As the angle of attack is increased to $22^{\circ }$ and $30^{\circ }$, the wake instability becomes sufficiently strong to overcome the stabilizing effect from the tip, resulting in periodic shedding of the wake vortices. Originated from the leading edge with predominately spanwise vorticity, the shedding vortices on the suction side gradually morph into the hairpin vortices with their legs extending far upstream. The tip vortex cannot be clearly observed due to its strong interaction with the unsteady shedding vortices. Similar vortical features have also been reported in the wake of a flat-plate wing at similar Reynolds numbers by Taira & Colonius (Reference Taira and Colonius2009).

For $sAR=2$, the increased spanwise extent alleviates the strong interaction between the tip vortex and unsteady shedding vortices as the case in $sAR=1$ wing. This enables the tip vortex to remain relatively steady despite the unsteadiness presented in the rest of the flow. The wake at $\unicode[STIX]{x1D6FC}=14^{\circ }$ exhibits weak periodic shedding according to a close examination of the lift force, although it may not be clearly visible in figure 9. At $\unicode[STIX]{x1D6FC}=22^{\circ }$, the unsteady shedding vortices mainly take the form of the braid-like structures, without clear spanwise vortex cores. As the angle of attack further increases to $\unicode[STIX]{x1D6FC}=30^{\circ }$, the unsteady vortex shedding grows stronger. Under its effect, the tip vortex exhibits noticeable undulations along the streamwise direction.

Increasing the aspect ratio to $sAR=4$ reveals the spanwise vortex shedding region. The spanwise vortical structures are quite weak at $\unicode[STIX]{x1D6FC}=14^{\circ }$ and appear only in the direct vicinity of the midspan. With the increase in angle of attack, the unsteady shedding vortices grow stronger and extend farther into the wake. The tip vortex elongates compared with that at lower angles of attack. At $\unicode[STIX]{x1D6FC}=30^{\circ }$, small-scale streamwise vortical structures emerge in the spanwise shedding region, making the wake complex. The streamwise undulations in the tip vortex become less significant compared with that in the case of $(\unicode[STIX]{x1D6FC},sAR)=(30^{\circ },2)$.

As we consider a higher aspect ratio of 6, the spanwise shedding takes place over an expanded region for $\unicode[STIX]{x1D6FC}=14^{\circ }$ and $22^{\circ }$. An additional feature that becomes clear at such high aspect ratio is the vortex dislocation, which divides the spanwise shedding region into two parts. This phenomenon is responsible for the low-frequency beating in the lift coefficients as shown in figure 2, and will be examined in detail in § 3.3. At $\unicode[STIX]{x1D6FC}=30^{\circ }$, smaller-scale streamwise vortical structures reminiscent of those induced by the Floquet instability (Deng, Sun & Shao Reference Deng, Sun and Shao2017; He et al. Reference He, Gioria, Pérez and Theofilis2017a) develop in the wake, making the unsteady shedding region even more complex.

Figure 10.

(a) Time-averaged streamwise vorticity fields at three streamwise locations of $x=2.5$, 5 and 7.5 for $(\unicode[STIX]{x1D6FC},sAR)=(22^{\circ },4)$. The three dashed lines are the contour lines of $\overline{\unicode[STIX]{x1D714}}_{x}=-1.0$, $-1.5$ and $-2.0$. (b) Circulations based on the three contour lines. (c) Length $L$ of the tip vortex defined based on the zero-crossing point of $|\unicode[STIX]{x1D6E4}_{x}|$–$x$ line ($\unicode[STIX]{x1D6E4}_{x}$ evaluated based on contour of $\overline{\unicode[STIX]{x1D714}}_{x}=-1.5$), and (d) decay rate $\text{d}|\unicode[STIX]{x1D6E4}_{x}|/\text{d}x$.

3.2.3 Characteristics of the tip vortex

We characterize the tip vortex in detail here for the representative case of $(\unicode[STIX]{x1D6FC},sAR)=(22^{\circ },4)$. The time-averaged streamwise vorticity fields in the tip region are shown in figure 10(a) at $x=2.5$, 5 and 7.5. The contours of $\overline{\unicode[STIX]{x1D714}}_{x}=-1.0$, $-1.5$ and $-2.0$ are shown to capture the shape of the vortex core. At $x=2.5$, the three contour lines are close to each other, depicting a mango-like shape. At $x=5$, the regions enclosed by the contours of $\overline{\unicode[STIX]{x1D714}}_{x}=-1.5$ and $-2.0$ shrink significantly. The vorticity level inside the vortex core also decreases. Further downstream at $x=7.5$, the tip vortex core becomes more diffused, and its shape relaxes to a circle.

The strength of the tip vortex can be quantitatively assessed through its streamwise circulation $\unicode[STIX]{x1D6E4}_{x}=\oint _{C}\boldsymbol{u}\boldsymbol{\cdot }\,\text{d}\boldsymbol{l}$, where the integration contour $C$ is taken as one of the three contours shown in figure 10(a). Evaluating the circulations along the streamwise direction, we obtain three $|\unicode[STIX]{x1D6E4}_{x}|$-$x$ profiles in figure 10(b). With increasing streamwise distance, the contour $C$ shrinks, and the vortex diffuses. Both factors contribute to the decay of the strength of tip vortices in a linear manner with increasing streamwise distance. Such linear decay of the tip vortices has also been observed in Edstrand et al. (Reference Edstrand, Sun, Schmid, Taira and Cattafesta2018b). The line associated with smaller contour level $|\overline{\unicode[STIX]{x1D714}}_{x}|$ measures extended length and reduced decay rate.

To characterize the strengths of the tip vortices, we use the contour of $\overline{\unicode[STIX]{x1D714}}_{x}=-1.5$ and calculate the length $L$ and the decay rate $\text{d}|\unicode[STIX]{x1D6E4}_{x}|/\text{d}x$ for different cases, as summarized in figures 10(c) and 10(d). We show in the Appendix that the choice of the contour level does not affect the insights drawn here. For the majority of the cases ($sAR\geqslant 2$, $\unicode[STIX]{x1D6FC}\leqslant 26^{\circ }$), the length of the tip vortex increases almost linearly with the angle of attack. The decay rate remains mostly constant at $\text{d}|\unicode[STIX]{x1D6E4}_{x}|/\text{d}x\approx 0.05$ due to viscous diffusion. Significant increase of the decay rate is observed at $\unicode[STIX]{x1D6FC}\approx 26^{\circ }{-}30^{\circ }$ with $sAR\gtrsim 2$ and $\unicode[STIX]{x1D6FC}\gtrsim 20^{\circ }$ with $sAR=1$. In these cases, the strong wake vortex shedding induces unsteady motion of the tip vortex, as observed in figure 9. This unsteadiness serves as an additional mechanism (to diffusion) to reduce the time-averaged strength of the tip vortex. As a result, the $L{-}\unicode[STIX]{x1D6FC}$ relationship as shown in figure 10(c) no longer follows the linear growth as at smaller $\unicode[STIX]{x1D6FC}$. At $sAR=1$ and 2, the time-averaged tip vortex length appears to even decrease at large angles of attack.

Figure 11.

(a) Spatial-temporal distribution of the cross-flow velocity probed at $(x,y)=(2,-0.3)$ for the case $(\unicode[STIX]{x1D6FC},sAR)=(22^{\circ },6)$. The grey arrow indicates the translating direction of the vortex dislocation. (b) Frequency distribution of the velocity over the spanwise direction. (c) Isosurfaces of $Q=1$ at three time instances showing one cycle of vortex dislocation. (d,e) The DMD (dynamic mode decomposition) modes ($Q$ criteria) for the two identified frequencies.

3.3 Spectral analysis of vortex dislocation

As discussed in § 3.2.2, for large-aspect-ratio wings, vortex dislocation occurs, dividing the spanwise vortex shedding region. Here, we focus on the particular case of $(\unicode[STIX]{x1D6FC},AR)=(22^{\circ },6)$. The cross-flow velocity $u_{y}$ along $(x,y,z)=(2,-0.3,0-7)$ over time is presented in figure 11(a). For $z\gtrsim 5$, which corresponds to the tip vortex region, the velocity remains steady over time. For $0\leqslant z\lesssim 5$, the variations in the velocity indicate the occurrence of the periodic vortex shedding. Along with the vortical structures in figure 9 for $(\unicode[STIX]{x1D6FC},sAR)=(22^{\circ },6)$, vortex dislocation is clearly observed from the discontinuities in the spanwise vortices. With the point of dislocation translating repeatedly from $z\approx 3.5$ to $z\approx 1$, the wake also changes over time, as observed from the isosurfaces of $Q=1$ in figure 11(c). At $t=95$, the spanwise vortical structures appear intact. Around $t=100$, vortex dislocation starts to develop in the near wake. By $t=105$, the formation of two shedding cells are clearly visible.

To gain additional insights, we apply a Fourier transform to the cross-flow velocity $u_{y}$ at each spanwise section, as shown in figure 11(b). We use $\hat{u} _{y}$ to denote the power spectral density (PSD) of the velocity fluctuation. The two spanwise shedding structures are found to be associated with different frequencies. The structures closer to the midspan shed with a frequency of $St_{1}=0.164$ ($St\equiv fc\sin \unicode[STIX]{x1D6FC}/U_{\infty }$, where $f$ is the dimensional shedding frequency). On the other hand, the structures closer to the wing tip shed with a lower frequency of $St_{2}=0.139$. For the current case, the point of dislocation translates along the wing span periodically from $z\approx 1$ to 3, at the beating frequency $St_{1}-St_{2}$. To identify the vortical structures associated with these two frequencies, we perform dynamic mode decomposition (Rowley et al. Reference Rowley, Mezić, Bagheri, Schlatter and Henningson2009; Schmid Reference Schmid2010) on the $Q$ criteria. As shown in figure 11(d), the modal structures corresponding to $St_{1}$ is composed of the columnar vortex cores aligned with the spanwise direction. In contrast, the mode with frequency $St_{2}$ presents three-dimensional modal structures comprised of skewed vortex cores as well as the streamwise vortical structures. These two modes overlap over $1\lesssim z\lesssim 3$. As time evolves, the time-varying phase between these two modes results in the changes in the vortical structures, as observed in figure 11(c).

Figure 12.

Frequency spectra of lift coefficients at $\unicode[STIX]{x1D6FC}=22^{\circ }$ for different aspect ratios. Here, ${\hat{C}}_{L}$ denotes the PSD of the lift coefficient. The PSD of the two-dimensional case is scaled down by a factor of 10 for visual clarity.

The dominant frequencies identified in the wake for $(\unicode[STIX]{x1D6FC},sAR)=(22^{\circ },6)$ are also detected in the corresponding lift spectrum, as shown in figure 12. Also overlaid are the lift spectra for other aspect ratios with $\unicode[STIX]{x1D6FC}=22^{\circ }$. The two-dimensional airfoil sheds vortices at the frequency of $St=0.174$, which is close to that reported by Rojratsirikul et al. (Reference Rojratsirikul, Genc, Wang and Gursul2011) and Calderon, Wang & Gursul (Reference Calderon, Wang and Gursul2013), regardless of the big difference in Reynolds numbers (the cited works cover $Re=10\,000{-}50\,000$). The dominant frequency $St_{1}$ for $sAR=6$ is close to the frequency in the two-dimensional case. This suggests the two-dimensional nature of the corresponding modal structure (mode 1) as shown in figure 11(d). On the other hand, $St_{2}$ almost coincides with the primary frequency in the cases of $sAR=2$ and 4, hinting that the lower shedding frequency is caused by the slowdown of shedding from the tip effects. The downward induced velocity from the tip vortex slows the shedding of the neighbouring vortical structures. For the much lower $sAR=1$ case, the strong interaction between the tip vortex and the unsteady vortices further slows down the vortex shedding, resulting in a lower shedding frequency. It should be noted that such a large influence of aspect ratio on Strouhal number is not observed at higher Reynolds numbers, at which the aspect ratio has a remarkably small effect on the shedding frequency, as reported by Rojratsirikul et al. (Reference Rojratsirikul, Genc, Wang and Gursul2011) and Calderon et al. (Reference Calderon, Wang and Gursul2013).

The lift spectra for different $sAR$ reveal how the tip effects influence the wake dynamics along the span. For $sAR\lesssim 4$, the tip effects dominate over the whole span of the wing, as discussed in §§ 3.2.1 and 3.2.2. For a higher aspect ratio, the tip effects become weaker over the extended region of $z\gtrsim 4$. This allows nominally two-dimensional structures to develop near the midspan and compete with the three-dimensional shedding near the tip, giving rise to vortex dislocation. Based on the insights obtained from spectral analysis, we are able to identify mainly three wake regions along the spanwise direction for large-aspect-ratio wings: (i) the tip vortex region as a direct consequence of the wing tip, (ii) the nominally two-dimensional shedding region near the midspan (mode 1) where the influence of the tip vortex is weak, and (iii) the three-dimensional shedding region (mode 2) that are influenced by the interaction between regions (i) and (ii).

3.4 Classification of the wake

We summarize our wake characterization over aspect ratio and angle of attack in figure 13. At $Re=400$, the wake remains stable for $\unicode[STIX]{x1D6FC}\lesssim 12^{\circ }$ for the aspect ratios considered in this study. For $sAR=1$, the flow can remain steady for even a higher angle of attack of $\unicode[STIX]{x1D6FC}=16^{\circ }$ due to the stabilizing tip effects. We shade in blue the conditions for a stable wake in figure 13. The transition from steady to unsteady flow is attributed to the Hopf bifurcation associated with the periodically shedding vortical structures (Ahuja et al. Reference Ahuja, Rowley, Kevrekidis, Wei, Colonius and Tadmor2007; Taira & Colonius Reference Taira and Colonius2009).

Figure 13.

Classification of the wake. ●, stable flow; ▴, steady tip vortex and unsteady shedding vortices; ▸, unsteady tip vortex; ▾, vortex dislocation.

With the increase in angle of attack, the vortex sheet generated from the leading and trailing edges form shedding vortices, as highlighted by the yellow region in figure 13. At this stage, the downwash provided by the tip vortex is no longer able to keep these wake vortices stationary. The wake here is characterized by the presence of two types of vortices: the unsteady shedding vortices and the tip vortex. The unsteady vortices take the form of spanwise vortex cores with their braid-like structures forming closed loops as described in § 3.2.1.

For higher-aspect-ratio wings, the unsteady wake exhibits an additional feature of vortex dislocation. With the tip vortex imposing downwash on the shedding vortices near the tip, their shedding frequency becomes lower than that of the shedding structures over the midspan region. When the discrepancy between these two frequencies become significant, the spanwise shedding processes become desynchronized, resulting in vortex dislocation as shown in figure 11(b). This phenomenon (shaded in purple) is observed for $sAR\gtrsim 4$ regardless of angle of attack, as long as $\unicode[STIX]{x1D6FC}$ is large enough to trigger vortex shedding. This type of vortex dislocation is ubiquitous for bluff body flows with spanwise inhomogeneity (Eisenlohr & Eckelmann Reference Eisenlohr and Eckelmann1989; Williamson Reference Williamson1989; Noack, Ohle & Eckelmann Reference Noack, Ohle and Eckelmann1991; Techet, Hover & Triantafyllou Reference Techet, Hover and Triantafyllou1998; Lucor, Imas & Karniadakis Reference Lucor, Imas and Karniadakis2001).

Figure 14.

Sectional (a) lift coefficients and (b) drag coefficients for $sAR=4$ with varying angles of attack. The thick curves represent the time-averaged values and the shaded regions represent their temporal fluctuations.

As we consider higher angles of attack, the tip vortex exhibits noticeable undulation in the streamwise direction due to its strong interaction with the unsteady shedding vortices. Such cases are observed over the red region in figure 13. We note that in realistic flows without the prescribed slip boundary condition at midspan, these flows might become asymmetric with respect to the midspan, as analysed in Taira & Colonius (Reference Taira and Colonius2009). We notice that the transition curve separating the unsteady tip vortex region from the yellow and purple regions is similar to the periodic/aperiodic transition in Taira & Colonius (Reference Taira and Colonius2009). This might suggest that the strong interaction between the tip vortices and the unsteady shedding vortices provides a route for the flow to transition to an asymmetric wake, if the symmetry plane is removed. Such transition is likely caused by the out-of-phase oscillations of the tip vortices at the two ends of the full wing. An unsteady tip vortex can also interact with the spanwise vortices experiencing vortex dislocation to create a complex wake as indicated by the top right subregion of figure 13. This complex wake generates finer-scale vortical structures from the rich interactions among the tip vortices and the large shedding vortices.

3.5 Aerodynamic forces

The three-dimensional wake imposes spanwise variations in the sectional distributions of the forces on the wing. The sectional lift and drag forces for $sAR=4$ are presented for various angles of attack in figure 14. For the cases of $\unicode[STIX]{x1D6FC}=8^{\circ }$ and $14^{\circ }$, $C_{l}$ gradually decreases from the midspan to the wing tip. The drag coefficient remains almost constant over the majority of the span except for the slight increase at the wing tip.

For angles of attack above $20^{\circ }$, considerable temporal fluctuations of $C_{l}$ appear in the unsteady shedding region, especially near $z\approx 2$. The temporal fluctuations in the drag coefficients are significantly lower than the lift coefficients. As the flow becomes steady towards the wing tip region, the temporal fluctuation of $C_{l}$ gradually disappears. We note here that the time-averaged sectional $\overline{C_{l}}$ exhibits a peak at $z\approx 3.5$ before it drastically drops at the wing tip. This is in contrast to the cases with lower angles of attack, where $C_{l}$ decays monotonically towards the tip. This difference in the lift profile will be related to the difference in the near-tip vortical structures in the discussion below.

Let us now resort to the force element theory (Chang Reference Chang1992; Lee et al. Reference Lee, Hsieh, Chang and Chu2012) to identify the wake structures that exert aerodynamic forces on the wing. In this theory, we utilize an auxiliary potential $\unicode[STIX]{x1D719}_{L}$ (take the lift force for example) that satisfies the boundary condition $-\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}_{L}=\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{e}_{\boldsymbol{y}}$. By taking the inner product of the Navier–Stokes equation (2.1b) with the potential velocity $\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}_{L}$ and integrating over the entire fluid domain $V$, the pressure force term $\int _{S}p\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{e}_{\boldsymbol{y}}\,\text{d}S$ in (2.2) can be expressed as a volume integration of wake variables and a surface integration. The lift force can then be expressed as

(3.2)$$\begin{eqnarray}F_{L}=\int _{V}\unicode[STIX]{x1D74E}\times \boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}_{L}\,\text{d}V+\frac{1}{Re}\int _{S}\unicode[STIX]{x1D74E}\times \boldsymbol{n}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}_{L}+\boldsymbol{e}_{\boldsymbol{y}})\,\text{d}S.\end{eqnarray}$$

The integrands in the first and second terms on the right-hand side are called the volume lift and the surface lift elements, respectively. The drag force elements can be obtained in a similar fashion by replacing the velocity potential $\unicode[STIX]{x1D719}_{L}$ with $\unicode[STIX]{x1D719}_{D}$ and enforcing $-\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}_{D}=\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{e}_{\boldsymbol{x}}$ on the wing surface.

Figure 15.

Volume force elements for $(\unicode[STIX]{x1D6FC},sAR)=(26^{\circ },4)$. (a) Lift elements at $t=90.17$ at which $C_{L}$ is the lowest. (b) Lift force element at $t=91.60$ at which $C_{L}$ is the highest. (c) Drag force element at $t=91.60$. In the top row, the isosurfaces of lift or drag force elements are shown with the contour level of $\pm 0.2$. The two-dimensional contour plots show the force elements at indicated $z$ planes with the black contours representing the vortex cores ($Q=1$). The magenta solid lines represent contours of positive force elements starting from $\unicode[STIX]{x1D74E}\times \boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}=1$ (outmost contour) with an increment of 2 towards the inside, and the cyan solid lines are the contours of negative force elements starting from $\unicode[STIX]{x1D74E}\times \boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}=-1$ (outmost) with an interval of $-2$.

Figure 16.

Isosurfaces of $\overline{Q}=3$ for an $sAR=4$ wing and varying angles of attack.

The lift elements for the case of $(\unicode[STIX]{x1D6FC},sAR)=(26^{\circ },4)$ at two time instances, at which the instantaneous lift coefficient is lowest and highest, are shown in figures 15(a) and (b), respectively. In the current study, the contribution of the viscous surface lift element is below 15 % of the total force. For this reason, we only report below the dominant volume lift element $\unicode[STIX]{x1D74E}\times \boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}_{L}$. Due to the spatially compact nature of the auxiliary potential gradient $\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}_{L}$, high levels of lift elements are concentrated in the vicinity of the wing. The two free shear layers emanating from both the suction and pressure sides contribute positively to lift, whereas the boundary layer on the pressure surface contributes adversely to the total lift. Similar observations have also been reported on an impulsively started finite-plate wing by Lee et al. (Reference Lee, Hsieh, Chang and Chu2012).

Comparing the snapshots between two time instances, the difference is observed in the unsteady shedding region. The difference in the lift elements between the two time instances at $z=1$ is subtle. However, at $z=2$, significant amounts of negative lift elements appear in figure 15(a) and disappear in figure 15(b). This explains the larger lift fluctuations at $z\approx 2$ than the other regions, as observed in figure 14(a).

Near the tip at $z=3.5$, the local flow is steady and the distributions of lift elements at two time instants are indistinguishable from each other. Noteworthy here is the large distribution of positive lift element near the wing tip giving rise to an increase of sectional lift at around $z=3.5$ in figure 14. Inward from the tip vortex is a vortical structure that forms by rolling up the leading-edge vortex sheet, which is seen from figure 16 for higher angles of attack. This also explains the absence of such local lift increase at lower angles of attack, as the rolled-up vortical structure from the leading edge does not form next to the tip vortex. Such local lift enhancement due to the downwash of tip vortices has also been discussed at length by DeVoria & Mohseni (Reference DeVoria and Mohseni2017) for steadily translating wings at high angles of attack.

Farther towards the tip at $z=3.9$, negative lift elements emerge in the near wake and the positive lift elements diminish, giving rise to the drastic drop in the sectional lift force. It is noteworthy that the trailing part of the tip vortex barely contributes to the aerodynamic forces. This is not only because of the fast decay of the velocity potential $\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}_{L}$, but also due to the fact that in the core of the tip vortex, the velocity vector and the vorticity vector are both aligned with the streamwise direction. The cross product $\unicode[STIX]{x1D74E}\times \boldsymbol{u}$ is small in magnitude, thus making the integrand in (3.2) negligible.

The volume drag elements for the same case are shown in figure 15(c). In general, the drag elements share similar patterns with that of the lift elements, where the two separated free shear layers contribute positively to the total drag and the boundary layer on the pressure side contributes negatively. The isosurfaces of drag elements, compared with the lift element at the same time instances, are devoid of the small-scale structures. Moreover, the intensity of the drag elements is lower compared to the lift elements, as judged from the absence of high-level contour lines of the drag elements. Thus, the integrated drag is smaller than lift in both temporal fluctuations as well as time-averaged value, as depicted in figure 14.

Figure 17.

Influence of aspect ratio on the sectional (a) lift and (b) drag coefficients at $\unicode[STIX]{x1D6FC}=22^{\circ }$. The force coefficients are aligned from the wing tip.

We further examine the effect of aspect ratio on the sectional lift coefficients for $\unicode[STIX]{x1D6FC}=22^{\circ }$ in figure 17. Here, the force coefficients are presented by aligning the distributions from the wing tip. With the increase in $sAR$, the lift coefficients increase, although they remain below the two-dimensional lift value at the same angle of attack. Focusing on the tip region, the peak in sectional lift near the tip is observed for finite-aspect-ratio wings. This is caused by the presence of the leading-edge vortical structure near the tip as discussed above. Such a structure gives rise to the increase in lift approximately 0.3–0.4 chords away from the tip. Similar to the lift coefficients, the drag coefficients increase with aspect ratio. However, the spanwise variations for the drag coefficients are relatively small compared to lift. The increase of the sectional $\overline{C_{d}}$ near the tip is also observed for all aspect ratios at the angle of attack of $\unicode[STIX]{x1D6FC}=22^{\circ }$.

Figure 18.

Time-averaged (a) lift coefficient, (b) drag coefficient and (c) lift-to-drag ratio.

At last, let us summarize our aerodynamic force data for the finite-aspect-ratio wings. The time-averaged lift coefficients $\overline{C_{L}}$, drag coefficients $\overline{C_{D}}$ and the lift-to-drag ratios $\overline{C_{L}/C_{D}}$ are plotted in figure 18. Overall, we notice that the lift and drag coefficients on the wing are smaller for lower-aspect-ratio wings and tend toward the two-dimensional limit for higher-aspect-ratio wings.

For the lift coefficients shown in figure 18(a), we observe that the tip effects reduce lift even at low angles of attack. For post-stall flows, the influence of the tip vortices on lift becomes more noticeable with wider variations in the lift values with respect to $sAR$. At a low Reynolds number of 400, lift coefficients monotonically increase for the range of angles of attack considered. The enhanced lift at higher angles of attack is created by the vortical structures that form above the wing as seen in figure 15 (Taira & Colonius Reference Taira and Colonius2009; Lee et al. Reference Lee, Hsieh, Chang and Chu2012). We can observe that above $\unicode[STIX]{x1D6FC}\approx 12^{\circ }$, there is a change in the lift slope which is caused by the emergence of unsteadiness in the wake, as shown in figure 13.

Next, let us examine the variation of drag coefficient over the angle of attack. As in the case with lift, we observe that the drag coefficients become higher as the aspect ratio increases. This is due to the flow being able to pass around the tip to reduce the drag coefficient for lower-aspect-ratio wings. However, we notice a striking feature that the drag coefficient is almost independent of the aspect ratio up to $\unicode[STIX]{x1D6FC}\approx 12^{\circ }$. This suggests that the drag force is primarily due to viscous effects even though the flow is separated at the moderate angles of attack. Once the wake becomes unsteady beyond $\unicode[STIX]{x1D6FC}\approx 12^{\circ }$, pressure effects contribute significantly to drag, which then is influenced by the aspect ratio and the resulting unsteady wake profiles. These observations are also in agreement with the findings for finite-aspect-ratio flat-plate wings from Taira & Colonius (Reference Taira and Colonius2009).

At this low Reynolds number, the lift-to-drag ratio is limited to $O(1)$, as evident from figure 18(c). Another unique feature for such low-$Re$ aerodynamics is the high angle of attack at which the peak lift-to-drag ratio is achieved. These ratios are generally in agreement with those for finite-$sAR$ flat-plate wings from (Taira & Colonius Reference Taira and Colonius2009), but achieve their peaks at slightly higher angles of attack for the NACA 0015 airfoil. Such observations are reported in Torres & Mueller (Reference Torres and Mueller2004) for a higher Reynolds number of $10^{5}$. If the finite wings are to be operated in steady condition, it can still achieve a high lift-to-drag ratio for $\unicode[STIX]{x1D6FC}\lesssim 12^{\circ }$. The ratio can be further increased but unsteadiness in the forces from the wake dynamics needs to be considered.