2 Experimental set-up

Planar digital particle image velocimetry (DPIV) measurements in the mid-span plane of the wings are performed in a recirculating wind tunnel that has a cross-section of $61\times 61~\text{cm}^{2}$ , a test section length of 2.44 m. Flat-plate, rectangular wing models of aspect ratios $\def\AR{A\mkern-8muR}\AR =0.75$ , 1, 1.5, 2, and 2.5 are investigated (see figure 1 a); the aspect ratio is conventionally defined as $\def\AR{A\mkern-8muR}\AR \equiv b_{max}^{2}/S=b/c$ , where $b_{max}\equiv b$ and $c$ are the rectangular wing span and chord. For each aspect ratio, the angle of attack is varied in $5^{\circ }$ increments from $0^{\circ }$ to $45^{\circ }$ ; the angle of attack is set manually to within $\pm 0.1^{\circ }$ using a digital protractor. The rectangular wings are machined from clear acrylic sheets; the chord is kept constant at $c=101.6$  mm and the span is varied to change the aspect ratio. The leading and trailing edges were rounded with elliptical profiles of ratios 5 : 1 (blunt) and 20 : 1 (sharp), respectively, while the side edges were left square. For $\def\AR{A\mkern-8muR}\AR\leqslant 2$ , the plate thickness is 3.175 mm, while for $\def\AR{A\mkern-8muR}\AR =2.5$ the thickness is 4.76 mm, giving thickness ratios of 3.1 % and 4.67 %, respectively. With a free stream velocity of $U=11.93~\text{m}~\text{s}^{-1}$ the chord-based Reynolds number is approximately $Re=80\,000$ .

The main objective of the current experiments is to study the effect of aspect ratio on the flow at the mid-span plane of the wings, and so the wing mounting device (figure 1 a) is designed to interfere as little as possible with this area. The device consists of two stainless steel rods (6.35 mm dia.) to which the plate is secured via countersunk machine screws. The distance between these holes is 50.4 mm and is centred about the mid-span of the plates; this spacing is the same for all $\def\AR{A\mkern-8muR}\AR$ so that the effect of sting on the mid-span flow remains the same. The sting rods may be rotated and extend back to a block housing, which can also rotate via its connection to a steel post through the tunnel ceiling panel. These two pivot points allow the angle of attack to be set while maintaining a relatively fixed position of the plate model in the camera image.

Figure 1.

(a) The wing cross-section, drawing of the mounting device and an image of the actual wings and mounting device. (b) The mean-normalized circulation standard deviation, $\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D6E4})/E(\unicode[STIX]{x1D6E4})$ , for each $\def\AR{A\mkern-8muR}\AR$ case. Inset: the time-dependent circulation, $\unicode[STIX]{x1D6E4}(t)$ , normalized by $E(\unicode[STIX]{x1D6E4})$ for a sample case. (c) Spatio-temporally averaged out-of-plane strain rate for each $\def\AR{A\mkern-8muR}\AR$ case to quantify three-dimensional effects. Inset: the spatial average of this strain rate as a function of time for a sample case.

The wind tunnel is seeded with olive oil particles ( ${\sim}1~\unicode[STIX]{x03BC}\text{m}$ ). The DPIV system consists of a high-speed CMOS 1 Mpx camera and a 20 mJ Nd : YLF laser. For each $\def\AR{A\mkern-8muR}\AR$ and $\unicode[STIX]{x1D6FC}$ combination, 400 image pairs are acquired at a frequency of $f_{s}=1000$  Hz. As an a posteriori check on this relatively short acquisition time of 0.4 s, circulations were computed from mean vorticity fields that were time averaged over an increasingly larger sample time; also computed were the corresponding standard deviations. It was found that major variations subsided by 0.2 s of sample time (i.e. 200 samples). Therefore, this suggests that the 0.4 s acquisition time is sufficient to capture the true time average of the vorticity fields and circulations, which are the main quantities reported in this paper.

Although the instantaneous flow is unsteady, these oscillations have an amplitude whose magnitude is acceptably smaller than that of the time-averaged flow. The inset of figure 1(b) plots the time-dependent positive circulation, $\unicode[STIX]{x1D6E4}(t)>0$ , normalized by its mean, $E(\unicode[STIX]{x1D6E4})$ , for a sample case. For this example, the circulation oscillates with amplitude approximately 10 % of the mean. The main plot of figure 1(b) shows the standard deviation of the circulation, again normalized by the mean, for each $\def\AR{A\mkern-8muR}\AR$ case as functions of the angle of attack. At small angles of attack this normalized standard deviation is larger because there is not much vorticity in the flow so that the mean itself is a small number (see figure 5). For $\unicode[STIX]{x1D6FC}>15^{\circ }$ the oscillation amplitudes for each case are again approximately 10 % of the corresponding mean. The instantaneous flow is also three-dimensional (see e.g. Yilmaz & Rockwell Reference Yilmaz and Rockwell2012), and we attempt to assess the degree of out-of-plane effects in the two-component, planar measurements. We compute the spatially averaged out-of-plane strain rate, $\unicode[STIX]{x2202}w/\unicode[STIX]{x2202}z=-(\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}x+\unicode[STIX]{x2202}v/\unicode[STIX]{x2202}y)$ , as a function of time. Large fluctuations due to spanwise flows are expected, but we assume that these fluctuate around a near-zero mean and therefore do not much affect the time-averaged two-dimensional flow fields. The inset of figure 1(c) shows this spatially averaged strain rate for a sample case, which is just after the static lift-stall angle. The inset shows fluctuations around a near-zero time average of $(\unicode[STIX]{x2202}w/\unicode[STIX]{x2202}z)_{avg}\approx 0.0037$ . The main plot in figure 1(c) shows the $(\unicode[STIX]{x2202}w/\unicode[STIX]{x2202}z)_{avg}$ for each $\def\AR{A\mkern-8muR}\AR$ as functions of angle of attack. For $\def\AR{A\mkern-8muR}\AR\geqslant 1.5$ , there are not drastic changes in the average strain rate over the range of $\unicode[STIX]{x1D6FC}$ , but for $\def\AR{A\mkern-8muR}\AR\leqslant 1$ the three-dimensional effects become increasingly significant. This is not too surprising since any tip vortex wandering (Kaplan, Altman & Ol Reference Kaplan, Altman and Ol2007) could pass into the mid-span plane. However, it is the square of the out-of-plane strain that is relevant to two-dimensionality (DeVoria & Ringuette Reference DeVoria and Ringuette2012), so the squared value should be small compared to relevant in-plane strains (e.g. vorticity/circulation), which is acceptably confirmed by the circulations plotted in figure 5.

The DPIV images are processed using an iterative multi-pass DPIV evaluation algorithm with window shifting/deformation. The rectangular interrogation windows start from $64\times 40~\text{px}^{2}$ and go to $32\times 20~\text{px}^{2}$ with 50 % overlap. The spatial resolutions of the DPIV data are then $\unicode[STIX]{x0394}x=0.029c$ and $\unicode[STIX]{x0394}y=0.018c$ in the horizontal and vertical directions, respectively. Measurement uncertainties are obtained by analysing images of the undisturbed free stream velocity and are estimated to be $\unicode[STIX]{x1D6FF}u/U\approx \unicode[STIX]{x1D6FF}v/U\approx 0.006$ . Using the smallest vector spacing, the uncertainty in the vorticity is $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D714}c/U=0.203$ , which is obtained from the ‘local circulation method’ (Raffel, Willert & Kompenhans Reference Raffel, Willert and Kompenhans1998).

Forces are measured using a specially designed micro-loading technologies (MLT) six-component internal force balance, which has been validated to measure small aerodynamic loads (Shields & Mohseni Reference Shields and Mohseni2013). Each model is swept through an angle of attack range from $\unicode[STIX]{x1D6FC}=0$ to $46^{\circ }$ in $2^{\circ }$ increments. At each $\unicode[STIX]{x1D6FC}$ , data acquisition is halted for 4 s to allow for initial flow transients to subside. Each force balance channel is sampled at 4096 Hz and a total of 16 384 samples are acquired. An identical sweep is made beforehand with the wind off, which acts as a tare set. The acquired data are then converted to aerodynamic loads using techniques outlined by the AIAA strain-gauge standard (AIAA Standards 2003). Blockage effects from streamline curvature, wake and solid bodies are corrected for based on methods presented by Rae & Pope (Reference Rae and Pope1984). The forces are non-dimensionalized into coefficients using $(\unicode[STIX]{x1D70C}U^{2}S)/2$ as a normalization.

3 Observed flow description

Before presenting the DPIV measurements, we introduce the lift and drag coefficients, $C_{L}$ and $C_{D}$ , in figure 2, which will aid the following discussions in relating changes of the flow character to the global quantities of lift and drag. Note that the static lift stall occurs at lower angles of attack as the aspect ratio increases. More specifically, the $\def\AR{A\mkern-8muR}\AR =0.75$ , 1, 1.5, 2 and 2.5 wings stall at $\unicode[STIX]{x1D6FC}_{s}\approx 40^{\circ }$ , $35^{\circ }$ , $22^{\circ }$ , $17^{\circ }$ and $12^{\circ }$ , respectively. As will be discussed below, this delay in lift stall correlates with decreased shedding of high-magnitude vorticity from the trailing edge.

Figure 2.

Direct force transducer measurements of the (a) lift coefficient $C_{L}$ and (b) drag coefficient $C_{D}$ as a function of angle of attack for $\def\AR{A\mkern-8muR}\AR =0.75$ , 1, 1.5, 2 and 2.5.

Note that since the flow fields are time averaged, changes in the angle of attack are regarded as parametric variations and are not to be confused with time-dependent variations that would occur during a dynamic manoeuvre. For $\unicode[STIX]{x1D6FC}<10^{\circ }$ the time-averaged flow fields for each $\def\AR{A\mkern-8muR}\AR$ are essentially the same and exhibit attached flow. The velocity field is nearly potential with vorticity being limited to a boundary-layer-like flow around the wing and shed into a thin wake at the trailing edge. The vorticity can here be described as a region of low total head and without definite separation of the flow from the wing (Goldstein Reference Goldstein1938), and as the incidence is increased this region approaches the leading edge. Steady, attached flow is the hallmark of classical aerodynamics and for this reason we do not discuss further the flow characteristics in this regime. Next, an observational discussion is provided on how the flow changes with incidence and aspect ratio.

Figure 3 plots the time-averaged vorticity fields for the cases of $\def\AR{A\mkern-8muR}\AR =0.75$ , 1, 1.5, 2 and 2.5 and for $\unicode[STIX]{x1D6FC}=10^{\circ }$ – $40^{\circ }$ . Also shown are the streamlines of the corresponding mean velocity fields. For the $\def\AR{A\mkern-8muR}\AR =0.75$ and 1 wings at $\unicode[STIX]{x1D6FC}=10^{\circ }$ the mean flow is still attached and maintains a potential-like velocity field with minor shedding of vorticity. For $\def\AR{A\mkern-8muR}\AR =1.5$ the magnitude of shed vorticity has increased, which is related to the boundary-layer instability growing larger in amplitude with intermittent separation, although this separation has not yet reached the leading edge. In the mean sense the flow is still nominally attached and no recirculating zone appears. For $\def\AR{A\mkern-8muR}\AR\geqslant 2$ strong shear layers appear close to or at the leading edge and are accompanied by reversed flow near the wing surface. The streamlines for these cases do not indicate a closed separation region. However, this behaviour is explained by examining movies of the instantaneous flow fields for $\def\AR{A\mkern-8muR}\AR\geqslant 2$ at $\unicode[STIX]{x1D6FC}=10^{\circ }$ , which show vortices close to the wing surface advecting downstream along a large portion of the chord, but lift off toward the trailing edge. Nonetheless, it can be concluded that decreasing the aspect ratio suppresses the growth of the boundary layer and delays the approach of separation toward the leading edge as $\unicode[STIX]{x1D6FC}$ is parametrically increased; this is similar to energizing the boundary layer.

Figure 3.

Time-averaged non-dimensional vorticity ( $\unicode[STIX]{x1D714}c/U$ ) fields and corresponding streamlines at mid-span for $\def\AR{A\mkern-8muR}\AR =0.75$ , 1, 1.5, 2 and 2.5 (left-to-right) and for $\unicode[STIX]{x1D6FC}=10^{\circ }{-}40^{\circ }$ (top-to-bottom). The colour bar in top left corner applies to all plots. Gaps apparent in the streamline data near the wing surface are due to the parallax view of the plate in the camera images.

By $\unicode[STIX]{x1D6FC}=15^{\circ }$ separation occurs at the leading edge for all $\def\AR{A\mkern-8muR}\AR$ cases tested, and a separation region exists, which is larger in size with increasing $\def\AR{A\mkern-8muR}\AR$ . For the purpose of discussion, we assume that the initial formation of the leading-edge shear layer results in a separation region, however small, with a stagnation/reattachment point, say $x_{r}$ , aft of the leading edge, and that this point approaches the trailing edge as $\unicode[STIX]{x1D6FC}$ is increased. Observation of figure 3 shows that the larger the aspect ratio the lower $\unicode[STIX]{x1D6FC}$ will be when $x_{r}$ reaches the trailing edge and results in drastic changes to the flow character, as can be verified by the differences from $\unicode[STIX]{x1D6FC}=10^{\circ }$ to $15^{\circ }$ in the flow fields of the $\def\AR{A\mkern-8muR}\AR =2$ and 2.5 wings. When $x_{r}$ passes the trailing edge it moves into the fluid and an opposite-sign shear layer is then generated at the trailing edge. The situation when $x_{r}$ has just passed the trailing edge is approximately seen for the $\def\AR{A\mkern-8muR}\AR =1.5$ wing at $\unicode[STIX]{x1D6FC}=15^{\circ }$ . An increase in $\unicode[STIX]{x1D6FC}$ or $\def\AR{A\mkern-8muR}\AR$ will exacerbate this situation and the reversed flow in the separation region becomes increasingly anti-parallel to the pressure side flow, thus increasing the strength of the trailing-edge vorticity. A counter-rotating separation region (CRSR) then forms and $x_{r}$ is at the rear of the separation region (e.g. $\def\AR{A\mkern-8muR}\AR\geqslant 1.5$ , $\unicode[STIX]{x1D6FC}=25^{\circ }$ cases).

The $\def\AR{A\mkern-8muR}\AR =0.75$ and 1 wings are unique in that they maintain reattached flow with a single separation region (i.e. the LESR), and thus continued lift generation, until $\unicode[STIX]{x1D6FC}_{s}\approx 35^{\circ }$ and $40^{\circ }$ , respectively. The peculiar source-like streamlines of the $\def\AR{A\mkern-8muR}\AR =0.75$ case at this high incidence are a result of increased three-dimensionality associated with the wandering of the tip vortices in and out of the mid-span plane (recall figure 1 c). A counter-rotating separation region is not observed for $\def\AR{A\mkern-8muR}\AR =0.75$ and 1 even at the highest angle of attack tested ( $\unicode[STIX]{x1D6FC}=45^{\circ }$ ), which is likely related to the tip vortex downwash impeding the roll-up process. However, a CRSR is certain to be present as $\unicode[STIX]{x1D6FC}\rightarrow 90^{\circ }$ , where the flow (in the time-averaged sense) is similar to bluff body or Prandtl–Batchelor flow (Batchelor Reference Batchelor1956; Saffman & Tanveer Reference Saffman and Tanveer1984a ).

For all aspect ratios, the drag obviously increases with angle of attack since the suction of the vortex flow (for either reattached or fully separated flow) generates a force in the direction of the normal of the wing surface (Polhamus Reference Polhamus1966), the horizontal component of which is $\sin \unicode[STIX]{x1D6FC}$ . Evidently, the drag penalty is then greater for the low- $\def\AR{A\mkern-8muR}\AR$ wings that maintain reattached flow (see figure 2).

Comparing the $C_{L}$ in figure 2 with the mean-flow fields in figure 3, it can be seen that the static lift stall occurs just after there is an increase of opposite-sign vorticity at the trailing edge. Therefore, the appearance of high-magnitude vorticity shedding at the trailing edge indicates the imminence of lift stall and a change in the flow structure. Reducing the aspect ratio (i.e. closer tip vortices) is seen to suppress the spatial growth of the separation region and to delay the approach of the reattachment point to the trailing edge. In fact, for the low- $\def\AR{A\mkern-8muR}\AR$ cases the weak, opposite-sign vorticity near the trailing edge at $\unicode[STIX]{x1D6FC}=15^{\circ }$ actually decreases and nearly vanishes as $\unicode[STIX]{x1D6FC}$ is increased further. Hence, it seems that the growing strength of the tip vortices with $\unicode[STIX]{x1D6FC}$ (Clark, Smith & Thompson Reference Clark, Smith and Thompson1975; Kaplan et al. Reference Kaplan, Altman and Ol2007) induces a flow that aids smoother merging of the flows from the suction and pressure sides of the wing at the trailing edge and consequently decreases shedding of strong trailing-edge vorticity.

4 Tip vortex downwash and reattachment

Since decreasing the $\def\AR{A\mkern-8muR}\AR$ brings the tip vortices in closer proximity of one another, the reattachment mechanism is often attributed to the momentum provided by their induced downwash (Winter Reference Winter1936; Jian & Ke-Qin Reference Jian and Ke-Qin2004; Taira & Colonius Reference Taira and Colonius2009b ). Similar explanations have been given for LEV attachment in wings performing dynamic manoeuvres, such as flapping wings using translation (Shyy et al. Reference Shyy, Trizila, Kang and Aono2009; Kim & Gharib Reference Kim and Gharib2010; Jardin et al. Reference Jardin, Farcy and David2012) and rotation (Birch & Dickinson Reference Birch and Dickinson2001; Ozen & Rockwell Reference Ozen and Rockwell2012). For the steadily translating wings studied here, this reasoning is very probable, however it remains speculative and does not offer further information on the details of the reattachment mechanism or the transition from reattached to fully separated flow as the aspect ratio or incidence is increased. Here, we use the volumetric reconstructions of DeVoria & Mohseni (Reference DeVoria and Mohseni2015) as well as our current measurements to provide more evidence of the tip vortex downwash as the reattachment mechanism.

DeVoria & Mohseni (Reference DeVoria and Mohseni2015) measured the three-dimensional flow around $\def\AR{A\mkern-8muR}\AR =1$ and 3 flat-plate, rectangular wings at $\unicode[STIX]{x1D6FC}=35^{\circ }$ with stereo digital particle image velocimetry. Figure 4 shows contour plots of the downward vertical velocity, $v/U<0$ , for these wings. For $\def\AR{A\mkern-8muR}\AR =1$ the flow is reattached and it is seen that $v/U$ has magnitudes in excess of half the free stream velocity. The plot locations shown are aft of the mid-chord location, where the LESR begins its reattachment (see figure 3), and so the large magnitudes of $v/U$ occur outside this region. This indicates that the tip vortices act mostly to reattach the downstream portion of the leading-edge shear layer onto the plate surface, rather than pinning the whole region to the wing. Conversely, for $\def\AR{A\mkern-8muR}\AR =3$ the flow is fully separated and the contours of $v/U$ are considerably weaker (near zero). More importantly, though, these regions of $v/U$ correspond to the counter-rotating recirculation zone within the massive separation region and are not representative of any tip vortex flow.

Figure 4.

Contour plots of the downward vertical velocity $v/U<0$ at several constant chord locations for (a)  $\def\AR{A\mkern-8muR}\AR =1$ , reattached flow, and (b)  $\def\AR{A\mkern-8muR}\AR =3$ , fully separated flow. Also shown are iso-surfaces of the negative spanwise vorticity shed ( $\unicode[STIX]{x1D714}_{z}c/U=-1$ ) from the trailing edge. These data are from the experiments of DeVoria & Mohseni (Reference DeVoria and Mohseni2015) for wings at $\unicode[STIX]{x1D6FC}=35^{\circ }$ and $Re=80\,000$ . (c) Integrated normalized vertical linear momentum versus $\unicode[STIX]{x1D6FC}$ for the data from the current paper for each $\def\AR{A\mkern-8muR}\AR$ .

One can imagine that the induced flow of the tip vortices has a significant effect on the flow near the trailing edge. Figure 4 also shows iso-surfaces of the negative spanwise vorticity component shed by the two wings, which is almost entirely generated from the trailing edge. For the $\def\AR{A\mkern-8muR}\AR =1$ case the trailing-edge vorticity is quickly wound up around the strong, well-formed tip vortices which are clearly identified or ‘tagged’ by the rolled up spanwise vorticity. On the other hand, for the $\def\AR{A\mkern-8muR}\AR =3$ case the trailing-edge vorticity extends well into the wake and the tip vortices have less effect on it, as already indicated in the previous paragraph. We reiterate that the trailing-edge vorticity shown in figure 4 corresponds to spanwise vorticity and not a streamwise vortex sheet as is usually described for conventional high-aspect-ratio wings at low incidences (e.g. Prandtl’s lifting line theory).

To provide further evidence, we use the data of the current experiments to estimate the total (i.e. integrated) linear momentum of the vertical velocity component, $v$ , within the measurement window. This is computed as $\int \unicode[STIX]{x1D70C}v\,\text{d}A$ and is normalized by $\unicode[STIX]{x1D70C}UA$ , which represents the linear momentum of the free stream integrated over the same measurement plane area $A$ . Also, recall that the chord and free stream speed are the same for each wing and so the only dimensional difference for a given angle of attack is the span. Imparting negative/downward vertical momentum into the fluid is, of course, indicative of lift generation. The right-most plot in figure 4 shows this integrated and normalized vertical linear momentum for each $\def\AR{A\mkern-8muR}\AR$ as a function of angle of attack. The lowest aspect ratios ( $\def\AR{A\mkern-8muR}\AR\leqslant 1$ ) show higher magnitudes of this momentum and a continued monotonic trend for a larger incidence range. The higher $\def\AR{A\mkern-8muR}\AR\text{s}$ do not continue to impart more downward momentum because either they cannot maintain reattached flow ( $\def\AR{A\mkern-8muR}\AR =1.5$ ) or the flow becomes fully separated at low incidences ( $\def\AR{A\mkern-8muR}\AR\geqslant 2$ , $\unicode[STIX]{x1D6FC}\approx 10^{\circ }$ ). Trends with similar physical implications are observed in the lift/drag coefficients (recall figure 2) and the mid-span circulation as will be shown in § 5.1.

Based on these quantitative results, it is quite reasonable to assume that for steadily translating low- $\def\AR{A\mkern-8muR}\AR$ wings the tip vortex downwash is indeed the mechanism that causes a reattached mean flow provided $\def\AR{A\mkern-8muR}\AR$ is sufficiently small, and that this mechanism is seemingly natural or passive.

5 Inviscid flow description

In this section, we interpret the observations of § 3 in a manner that is consistent with inviscid, potential flow concepts. For this purpose, we first regard the mean flow in the mid-span plane as two-dimensional with the three-dimensional effects of a finite aspect ratio and the tip vortex flow represented as some external agency that can alter the flow character. First, recall that potential flow theory allows a solid body in a uniform flow to be represented by the closed streamline corresponding to the complex potential obtained from the superposition of the potentials for a vortex doublet (or source doublet) and a plane flow. The Kutta condition is then imposed by the development of the bound circulation $\unicode[STIX]{x1D6E4}_{b}$ around the wing, with the Helmholtz law of persistence of irrotationality being satisfied by the ‘starting vortex’ of equal and opposite circulation $\unicode[STIX]{x1D6E4}_{s}$ located at an infinite distance downstream.

Consider attached flow and allow the wing to move impulsively from rest to its steady translation speed. The bound circulation will build up to the magnitude of the starting vortex. The effect of the starting vortex on the build-up of $\unicode[STIX]{x1D6E4}_{b}$ is often referred to as the Wagner effect. For finite- $\def\AR{A\mkern-8muR}\AR$ wings, the induced velocity of the tip vortex flow results in a similar effect. However, since the tip vortices move with the wing, this effect is omnipresent meaning that the steady-state bound circulation value for a low- $\def\AR{A\mkern-8muR}\AR$ wing is less and consequently less lift is generated as compared to a higher- $\def\AR{A\mkern-8muR}\AR$ wing at the same angle of attack in the attached flow regime (see figure 2). This is, of course, consistent with the effective angle of attack concept associated with lifting line theory and which also results in reduced lift.

Now, at some angle of attack the bound circulation will be too large to remain completely bound to the wing. Physically, this corresponds to separation and a leading-edge vortex sheet is released into the fluid. In the case of reattached flow, a single LESR forms and has a rotation that is of the same sense as the bound circulation. The irrotational flow is now that around the closed streamline of the LESR (similar to Saffman and coworkers), and around which there is an effective bound circulation that includes the vortex sheet strength, say $\unicode[STIX]{x1D6E4}_{le}$ . Furthermore, the vortex sheet must have an ‘image system’ inside the wing in order to satisfy the no-flow-through condition. The Helmholtz law becomes

(5.1) $$\begin{eqnarray}(\unicode[STIX]{x1D6E4}_{b}+\unicode[STIX]{x1D6E4}_{le})-\unicode[STIX]{x1D6E4}_{s}=0.\end{eqnarray}$$

Here, the reattachment point $x_{r}$ exists on the wing and thus the flow over the trailing edge is able to merge smoothly with the flow coming from the pressure side, thus establishing a Kutta condition at the trailing edge. So long as the flow stays reattached, the effective bound circulation, $\unicode[STIX]{x1D6E4}_{o}=\unicode[STIX]{x1D6E4}_{b}+\unicode[STIX]{x1D6E4}_{le}$ , results in lift enhancement.

Similar to the way in which the bound circulation cannot grow indefinitely in a real viscous fluid, so too must the effective bound circulation of the reattached flow be limited. Stated differently, as the angle of attack and/or $\def\AR{A\mkern-8muR}\AR$ increases the image system of the leading-edge vortex sheet will, at some point, no longer be able to stay confined inside the wing. The ‘excess’ image vorticity is then released into the flow via shedding of an opposite-sign vortex sheet at the trailing edge. This occurs when the reattachment point reaches the trailing edge (recall the discussions of § 3). Representing the trailing-edge vortex sheet by $\unicode[STIX]{x1D6E4}_{te}$ , the Helmholtz law now yields

(5.2) $$\begin{eqnarray}(\unicode[STIX]{x1D6E4}_{b}+\unicode[STIX]{x1D6E4}_{le}-\unicode[STIX]{x1D6E4}_{te})-\unicode[STIX]{x1D6E4}_{s}=0.\end{eqnarray}$$

The trailing-edge vortex sheet contribution, $-\unicode[STIX]{x1D6E4}_{te}$ , to the effective bound circulation, $\unicode[STIX]{x1D6E4}_{o}=\unicode[STIX]{x1D6E4}_{b}+\unicode[STIX]{x1D6E4}_{le}-\unicode[STIX]{x1D6E4}_{te}$ , thus results in a negative contribution to the lift. The argument that lift loss occurs when $x_{r}$ passes the trailing edge garners support from the solutions given by Saffman & Sheffield (Reference Saffman and Sheffield1977) for a standing point vortex over a wing. They enforced the condition that the (finite) trailing-edge velocity tangential to the wing be in the downstream direction. It was found that maximum lift occurs when the trailing edge is a stagnation point, i.e. when $x_{r}$ exists there. More solutions exist, but which violate the imposed condition meaning that there is reversed flow over the trailing edge. This reversed flow is exactly the situation described in § 3 that creates trailing-edge vorticity and leads to lift stall. Hence, in the context of the potential solutions, an opposite-sign point vortex, namely $-\unicode[STIX]{x1D6E4}_{te}$ , should be introduced to represent the different physics. More concrete, experimental evidence of lift loss coinciding with $x_{r}$ reaching the trailing edge was shown by Rival et al. (Reference Rival, Kriegseis, Schuab, Widmann and Tropea2014), where the instantaneous lift of a plunging two-dimensional profile dropped rapidly as the LEV detachment process progressed. We also further quantify and discuss this concept with our experimental measurements of circulation in § 5.1.

It is also interesting to comment on the starting vortex as a source of the image system vorticity. If we let the single separation region be represented by a free vortex $\unicode[STIX]{x1D6E4}_{le}$ , then the image system consists of a same-sign vortex $\unicode[STIX]{x1D6E4}_{le}$ at the origin and an opposite-sign vortex $-\unicode[STIX]{x1D6E4}_{le}$ at the inverse point inside the cylinder. The bound circulation is also located at the origin. Hence, with (5.1) the vortices at the origin could be thought of as the mirror image of the starting vortex, which is located at infinity. The image system is then decomposed into the mirror images of the starting vortex and the free vortex. When the trailing-edge vortex sheet is shed, (5.2) indicates that $\unicode[STIX]{x1D6E4}_{s}$ will decrease as the trailing-edge vorticity lowers the effective bound circulation of the counter-rotating separation region. Recognizing that the Joukowski transformation maps the point at infinity in the physical plane to the origin of the virtual circle plane, we can then view this decrease of $\unicode[STIX]{x1D6E4}_{s}$ as circulation returning to the wing, which then exits again at the trailing edge. Accordingly, at very high angles of attack the CRSR becomes more symmetrical and $\unicode[STIX]{x1D6E4}_{s}$ continually decreases. Then as $\unicode[STIX]{x1D6FC}\rightarrow 90^{\circ }$ no starting vortex is shed, $\unicode[STIX]{x1D6E4}_{s}=0$ , and assuming symmetry of the flow then $\unicode[STIX]{x1D6E4}_{le}=\unicode[STIX]{x1D6E4}_{te}$ and so $\unicode[STIX]{x1D6E4}_{b}=0$ .

5.1 Experimental circulation

Here, we use the measurements to quantitatively validate the statements made previously about the loss of lift occurring when the reattachment point passes the trailing edge and vorticity is generated there. Again, Rival et al. (Reference Rival, Kriegseis, Schuab, Widmann and Tropea2014) observed this to be the case for a dynamic manoeuvre of two-dimensional wings. With the experimental data we cannot distinguish a bound circulation, however the positive and negative contributions to the total circulation can be measured. For simplicity, let the regions of total positive and negative vorticity in the flow be represented by separate circulation magnitudes: $\unicode[STIX]{x1D6E4}^{+}$ and $\unicode[STIX]{x1D6E4}^{-}$ , respectively. Then, the Joukowski lift formula gives:

(5.3) $$\begin{eqnarray}L=\unicode[STIX]{x1D70C}U(\unicode[STIX]{x1D6E4}^{+}-\unicode[STIX]{x1D6E4}^{-}).\end{eqnarray}$$

When there is no separation $\unicode[STIX]{x1D6E4}^{+}$ is essentially the bound circulation of classical steady aerodynamics. For separated, but reattached flow $\unicode[STIX]{x1D6E4}^{+}\gg \unicode[STIX]{x1D6E4}^{-}$ and the lift is enhanced for a given $\def\AR{A\mkern-8muR}\AR$ by the separation region. Finally, for fully separated flow the strength of the trailing-edge vorticity increases significantly as discussed previously and if the increase of $\unicode[STIX]{x1D6E4}^{+}$ with a corresponding increase of $\unicode[STIX]{x1D6FC}$ is less, then the lift will decrease. Figure 5 plots the non-dimensional experimental $\unicode[STIX]{x1D6E4}^{+}$ and $\unicode[STIX]{x1D6E4}^{-}$ as functions of $\unicode[STIX]{x1D6FC}$ for the different aspect ratios. The circulations are computed from the (discrete) area integral of vorticity of the positive and negative vorticity fields, which are thresholded above the measurement uncertainty level to avoid accumulation of low-level vorticity in the result. It is seen that $\unicode[STIX]{x1D6E4}^{+}$ generally grows with $\unicode[STIX]{x1D6FC}$ and shows a marked increase when the leading-edge shear layer forms ( $\unicode[STIX]{x1D6FC}\approx 5$ – $10^{\circ }$ ). For $\def\AR{A\mkern-8muR}\AR\geqslant 1.5$ , the same is true for $\unicode[STIX]{x1D6E4}^{-}$ . However, for $\def\AR{A\mkern-8muR}\AR\leqslant 1$ , $\unicode[STIX]{x1D6E4}^{-}$ is seen to decrease in magnitude after the formation of the leading-edge shear layer, which explains why the decrease of trailing-edge vorticity via tip vortex downwash for low- $\def\AR{A\mkern-8muR}\AR$ wings results in continued lift generation at large angles of attack.

Figure 5.

The measured non-dimensional circulation magnitudes associated with (a) positive vorticity $\unicode[STIX]{x1D6E4}^{+}/Uc$ , and (b) negative vorticity $\unicode[STIX]{x1D6E4}^{-}/Uc$ as a function of angle of attack for each aspect ratio. Note that at the largest values of $\unicode[STIX]{x1D6FC}$ the vorticity field of a few $\def\AR{A\mkern-8muR}\AR$ cases is not entirely contained within the measurement plane (figure 3). However, the missing vorticity is of low magnitude.

In fact, Taira & Colonius (Reference Taira and Colonius2009a ) found that flow control via downstream blowing along the trailing edge was most effective in maintaining increased lift at larger incidences for low- $\def\AR{A\mkern-8muR}\AR$ wings at $Re=300$ . They concluded that the downstream blowing at the trailing edge increased the strength/downwash of the tip vortices such that the leading-edge vortices were pinned to the wing. Mejia et al. (Reference Mejia, Moser, Brzozowski and Glezer2011) showed similar increases in the lift of a NACA 4415 airfoil at $Re=900\,000$ using synthetic jet actuation at the trailing edge. In the context of this paper, the downstream blowing/jetting are analogous to decreasing the aspect ratio and the concomitant increase in the influence of the tip-vortex-induced flow facilitates a smooth transition at the trailing edge, thus eliminating or mitigating the generation of trailing-edge vorticity.

6 Topological flow description

In this section we show that, from a topological point of view, the counter-rotating separation region is always present regardless of the particular physical flow configuration, and can be interpreted as a vortex pair. First, we introduce the relation given by Hunt et al. (Reference Hunt, Abell, Peterka and Woo1978) that is satisfied by the critical/fixed points within a two-dimensional plane section of the flow (e.g. the mid-span plane):

(6.1) $$\begin{eqnarray}\left(\sum N+\frac{1}{2}\sum N^{\prime }\right)-\left(\sum S+\frac{1}{2}\sum S^{\prime }\right)=1-n,\end{eqnarray}$$

where $N$ and $N^{\prime }$ are full and half-nodes, $S$ and $S^{\prime }$ are full and half-saddles and $n$ is the connectivity of the plane, which is $n=2$ for the current case; $N$ , $N^{\prime }$ and $S$ , $S^{\prime }$ are also recognized as four-way/three-way elliptical and hyperbolic critical points, respectively (Lipinski, Cardwell & Mohseni Reference Lipinski, Cardwell and Mohseni2008). In fully attached flow, the only critical points are two half-saddles at the leading and trailing edges. Alternatively, the vorticity can be thought of as entirely confined in the wing as a vortex doublet and each sign of vorticity represents the boundary layers on the wing. The streamlines of a vortex pair are shown in figure 6(a), and there are two full nodes and two full saddles. However, when the body streamline is mapped to the flat-plate wing, the two nodes cancel and the saddles reduce to half-saddles to be consistent with (6.1).

Figure 6.

(a) Streamlines of a vortex pair; grey squares/circles are nodes/saddles. (b) Separated, but reattached flow with $x_{r}$ existing on the wing. The separation region has an image inside the wing. (c) Just after $x_{r}$ has passed the trailing edge and the excess image vorticity has begun to shed into the fluid. (d) Fully separated flow, where the forming counter-rotating separation region is fed by the image vorticity. Despite the different physical ramifications of these cases, the vortex structures can be thought of as topologically equivalent.

The situation when the flow is separated at the leading edge, but reattaches is shown in figure 6(b). The forward and rear stagnation points of the LESR are also half-saddles. Alternatively, the anti-clockwise half of the vortex pair is allowed to enter the fluid as a full node via the leading-edge singularity and represents the LESR; the clockwise half remains confined to the wing as part of the image system. For the time-averaged flows here, the forward stagnation point always coincides with the leading edge and thus a full saddle exists there, consistent with the full saddle of the vortex pair. Up to this point, the mean-flow topology is similar to that described by Rival et al. (Reference Rival, Kriegseis, Schuab, Widmann and Tropea2014) for the LEV of two-dimensional plunging profiles. However, we observe different characteristics of the mean flow when $x_{r}$ moves past the trailing edge and into the fluid.

Figure 6(c) shows the case when $x_{r}$ has just passed the trailing edge. The excess image vorticity is shed to create the trailing-edge vortex sheet and the clockwise half of the vortex pair enters the fluid as a full node through the trailing-edge singularity. The flow does not immediately go around the trailing edge, but rather continues along a dividing streamline coincident with the vortex sheet, and so either a full or half-saddle exists at the trailing edge. When $x_{r}$ goes further beyond the trailing edge (figure 6 d), the image vorticity is continually fed into the fluid at the trailing-edge singularity and the counter-rotating separation region grows in size. Figure 3 confirms these flow patterns. Now, the CRSR configuration must also be consistent with (6.1). So far, there are two full nodes representing the counter-rotation, and two full saddles at the leading edge and at the rear of the separation region. The recirculating flow of the trailing-edge node connects back to the plate as a half-saddle, and hence there must be a half-saddle at the trailing edge. However, figure 3 indicates that this flow connects back right at the trailing edge so that the two half-saddles merge into a full saddle, which is consistent with the trailing edge as the source of vorticity. Hence, the topology of the flow can always be viewed as a vortex pair with additional half-saddles to account for physical flow separations.

7 Concluding remarks

The mean-flow reattachment of steadily translating low-aspect-ratio wings ( $\def\AR{A\mkern-8muR}\AR =0.75{-}2.5$ ) was investigated experimentally with planar DPIV at the mid-span plane. The chord-based Reynolds number was $Re=80\,000$ and the angle of attack was ranged from zero to $45^{\circ }$ for each wing. Continued lift generation at high incidences is achieved only by wings with $\def\AR{A\mkern-8muR}\AR <1.5$ , because reattached mean flow is maintained by the tip vortex downwash, whereas fully separated flow occurs for larger aspect ratios.

The tip-vortex-induced flow facilitates a smooth flow at the trailing edge and therefore helps enforce a Kutta condition there. As a result, the strength of vorticity shed at the trailing edge is decreased, and this consequently results in the continued generation of Joukowski-type lift at high angles of attack for sufficiently low aspect ratios. Just below the static lift-stall angle of attack, strong vorticity is shed at the trailing edge, meaning that the reattachment becomes increasingly intermittent and will lead to large fluctuations in the lift force that are characteristic of incidences near stall. This suggests that with a further increase of incidence or aspect ratio the tip vortex downwash will become even less effective. As such, the intermittent reattachment and eventual fully separated flow will spread across the trailing edge of the wing leading to stall. Hence, the appearance of strong trailing-edge vorticity at mid-span could serve as stall detection.

The flow was also described through concepts consistent with inviscid, potential flow. Reattached flow is characterized by an effective bound circulation around the wing and separation region, resulting in the continued lift generation. When vorticity is shed from the trailing edge, the effective bound circulation is decreased, leading to lift stall. These statements were confirmed with the experimental measurements of mid-span circulation. Lastly, the flow was described with a topological analysis. The mean-flow topology can be interpreted as a vortex pair with 0, 1 and 2 additional half-saddles representing the attached, reattached and fully separated physical flow configurations, respectively.

The exact angle of attack and value of the aspect ratio for which the flow can be reattached at high incidences is also dependent on the Reynolds number, meaning the reattachment process can be assisted by transition to turbulence in the leading-edge shear layer. Nonetheless, we believe that the general progression of the vortex topology as described in this paper can be extended or generalized to other flow regimes. Reynolds number effects as well as different planform and leading-edge geometries are ideal subjects for future work in this area. Also important is the unsteady vortex shedding and its connection to the instantaneous aerodynamic loads on low- $\def\AR{A\mkern-8muR}\AR$ wings and is a topic currently under investigation by the authors.