1. Introduction
In nature, the excellent flying and swimming patterns of animals are mostly achieved via flapping wings or fins (Azuma Reference Azuma2012). The geometries of these fins and wings vary among species, while they mostly share a common feature, i.e. a swept shape with a nonlinear leading-edge curve with respect to the span. Figure 1 sketches several wing and fin examples with their leading edges outlined by red lines. A gliding swift can morph its wings to a lunate profile with the sweep-back angle (that will be noted as the sweep angle in this work) of the hand wing up to 60
$^{\circ }$
(Videler Reference Videler2006). Aerodynamic research proved that this swept profile is superior for fast gliding and steering since it helps to bear extreme loads during these manoeuvres (Lentink et al. Reference Lentink2007). The swept morphing also occurs in bird perching, during which the flapping wing undertakes a concomitant pitch-up to augment lift and enable a smooth deceleration until touchdown (KleinHeerenbrink et al. Reference KleinHeerenbrink, France, Brighton and Taylor2022). For tiny flyers, such as cicadas, their wing geometries are mostly maintained during flights (Song et al. Reference Song, Lee, Soh, Zhu and Bai2004). Due to the reinforcement of strong veins, the leading edges of their wings always follow a lunate shape. Moreover, flapping fins also take shape as a lunate plate, e.g. the tuna fin, which has been demonstrated to reduce the incidence of dynamic stall and thus extend the propulsion performance envelope to higher angles of attack (Smits Reference Smits2019).
Wings or fins with nonlinear leading-edge profiles.
The aerodynamic profit of these wings and fins is associated with a well-known fluid phenomenon, i.e. the roll-up of the leading-edge shear layer and thus the formation of the leading-edge vortex (LEV). Since the first demonstration of Ellington and his colleagues (Ellington et al. Reference Ellington, Van Den, Coen, Alexander and Thomas1996; Van Den Berg & Ellington Reference Van Den Berg and Ellington1997), the LEV has been found to be continuously attached to the dorsal side of a flapping wing and fin during a stroke, and its detachment happens at the stroke reversal. This encouraged a further extension of the dynamic stall mechanism (Ellington Reference Ellington1984; Dickinson & Götz Reference Dickinson and Götz1993), which, before these works, was adapted from unsteady airfoil aerodynamics to explain the high aerodynamic forces on flapping wings and fins. The absence of stall mechanism was thus proposed to emphasise the prolonged LEV attachment during a stroke. Numerous studies have been conducted to explain the stabilising mechanisms that sustain LEV attachment, since otherwise, it should shed from the wing in the process of a stroke. This has been observed on translating wings with either high angles of attack or rapid pitch-up manoeuvres (Dickinson & Götz Reference Dickinson and Götz1993; Chen, Colonius & Taira Reference Chen, Colonius and Taira2010; Yilmaz & Rockwell Reference Yilmaz and Rockwell2012; Hord & Lian Reference Hord and Lian2016; Chen et al. Reference Chen, Zhang, Zhou and Wu2023b ; Chen & Wu Reference Chen and Wu2024).
Compared with translating wings, the prolonged attachment of LEV on flapping wings and fins is believed to originate from the revolving nature of the motion (Chen et al. Reference Chen, Cheng and Wu2023a
). In the pioneering research of Ellington et al. (Reference Ellington, Van Den, Coen, Alexander and Thomas1996), they indicated that the flapping motion induces a strong axial flow inside the LEV core and benefits the LEV stability by transporting the accumulated vorticity towards the wing tip. This axial flow can be established via three effects, i.e. the dynamic spanwise pressure gradient, the centrifugal acceleration in the boundary layer and the induced velocity of a spiral vortex line. Despite the fact that many following studies have evidenced a prominent axial flow and thus a strong helical LEV (Lan & Sun Reference Lan and Sun2001; Pick & Lehmann Reference Pick and Lehmann2009; Jardin, Farcy & David Reference Jardin, Farcy and David2012; Ozen & Rockwell Reference Ozen and Rockwell2012; Carr, Chen & Ringuette Reference Carr, Chen and Ringuette2013; Garmann, Visbal & Orkwis Reference Garmann, Visbal and Orkwis2013), novel mechanisms may be involved in the stable LEV at Reynolds numbers (
$Re$
) near 100 because the axial flow becomes negligible (Birch & Dickinson Reference Birch and Dickinson2001; Shyy & Liu Reference Shyy and Liu2007; Cheng et al. Reference Cheng, Sane, Barbera, Troolin, Strand and Deng2013). Another explanation relates the LEV stability to Rossby number (Ro) or aspect ratio (AR), which mathematically scales the strength of centripetal and Coriolis accelerations (Lentink & Dickinson Reference Lentink and Dickinson2009a
), and a stable LEV is maintained by strong rotational accelerations. Thus, the stable LEV is limited within a certain spanwise range (Lentink & Dickinson Reference Lentink and Dickinson2009b
; Kruyt et al. Reference Kruyt, Van, Gertjan, Altshuler and Lentink2015; Jardin & Colonius Reference Jardin and Colonius2018). This motivated a new branch of research that either examines the role of rotational accelerations in maintaining LEV stability (Garmann & Visbal Reference Garmann and Visbal2014; Jardin Reference Jardin2017; Werner et al. Reference Werner, Chung, Wang, Liu, Cimbala, Dong and Cheng2019) or determines the optimal AR (Harbig, Sheridan & Thompson Reference Harbig, Sheridan and Thompson2013; Phillips, Knowles & Bomphrey Reference Phillips, Knowles and Bomphrey2015; Lee, Lua & Lim Reference Lee, Lua and Lim2016).
Besides, the downwash effects and the mixing of opposite-signed boundary layer vorticity are also argued to contribute to LEV stability (Birch & Dickinson Reference Birch and Dickinson2001; Wojcik & Buchholz Reference Wojcik and Buchholz2014). Recently, Chen and his colleagues revisited the spatio-temporal LEV evolution on a revolving wing from a vorticity transport perspective (Chen, Wu & Cheng Reference Chen, Wu and Cheng2019, Reference Chen, Wu and Cheng2020a
, Reference Chen, Wang, Zhou, Wu and Cheng2022, Reference Chen, Zhou, Werner, Cheng and Wu2023c
). The relative importance of the spanwise convection, the downwash effect, the vortex tilting/stretching (due to the revolving nature) and the diffusion at various
$Re$
, AR and Ro were discussed. A novel mechanism, i.e. the dual-stage radial–tangential vortex tilting, was proposed to further demonstrate the contribution of vortex tilting in stabilising the LEV, and this mechanism first shows the feedback feature in vorticity regulation (Chen et al. Reference Chen, Zhou, Werner, Cheng and Wu2023c
).
As reviewed by Chen et al. (Reference Chen, Cheng and Wu2023a
), most works on LEV stability and high lift generation are based on simplified rectangular plates with straight leading edges against the incoming flow. The impact of the nonlinear leading-edge profile was assumed to be trivial and thus ignored. However, according to an examination of LEV bursting near the distal edge of the wing (Jones et al. Reference Jones, Medina, Spooner and Mulleners2016; Medina & Jones Reference Medina and Jones2016), the local expansion of LEV size is associated with a reversed spanwise flow and a weaker pressure gradient. This recalls the traditional LEV bursting near the trailing edge of delta wings (Gursul, Wang & Vardaki Reference Gursul, Wang and Vardaki2007), in which the loss of swirl level and pressure gradient initiates the vortex bursting. For delta wings, increasing the sweep angle (
$\beta$
) can enlarge the spanwise projection of the incoming velocity (figure 2), which helps to maintain the axial flow and pressure gradient, therefore extending the stable vortex region. These similarities inform us that a properly swept leading-edge profile may also improve the LEV stability on flapping wings and fins by manipulating the spanwise flow.
Incoming flow decomposition on a nominal swept wing.
Inspired by this hypothesis, Lentink & Dickinson (Reference Lentink and Dickinson2009b
) compared the flow of translating and revolving insect wings at various sweep angles. They found that translating wings (both swept and unswept) rapidly shed their LEV at
$Re$
= 110 and 1400, while revolving wings retained a stable LEV at
$Re$
= 110. Thus, they concluded that the mechanisms for vortex stabilisation on translating delta wings may not appear in insect wings at
$Re$
= 110. This was supported by the following research on parallelogrammical swept wings in either translating or plunging motions (Lim et al. Reference Lim, Teo, Lua and Yeo2009; Beem, Rival & Triantafyllou Reference Beem, Rival and Triantafyllou2012; Wong, Kriegseis & Rival Reference Wong, Kriegseis and Rival2013). Specifically, Wong et al. (Reference Wong, Kriegseis and Rival2013) proved that the spanwise flow enhancement of swept wings can only regulate the LEV strength when coupled with a spanwise gradient of the effective incidence. This explained the limited benefits of swept wings when they are subject to translating or plunging motions, in which the effective incidence is fixed along the span. In contrast, the enhanced spanwise flow of swept wings was able to improve the LEV stability during pitching (Hartloper & Rival Reference Hartloper and Rival2013; Onoue & Breuer Reference Onoue and Breuer2017; Zangeneh Reference Zangeneh2021), flapping (Wong & Rival Reference Wong and Rival2015) and perching kinematics (Adhikari et al. Reference Adhikari, Loubimov, Kinzel and Bhattacharya2022). However, a recent numerical study presented an exception, in which varying
$\beta$
only causes trivial changes to LEV structures on round-tip wings under plunging and flapping kinematics (Zurman-Nasution, Ganapathisubramani & Weymouth Reference Zurman-Nasution, Ganapathisubramani and Weymouth2021). This contradictory conclusion is supported by 39 simulations with AR = 2–8 and
$\beta$
= 20
$^{\circ }$
–40
$^{\circ }$
, but less information on spanwise flow and vorticity transport was reported.
Besides the role of the spanwise gradient of effective incidence, another debate is the specific vorticity transport that contributes to LEV stability on swept wings. According to the vorticity transport (1.1,
$\tau$
is the dimensionless time,
$\boldsymbol{\omega }$
is the vorticity, and
$\boldsymbol{u}$
is the velocity), the strength of spanwise flow determines the spanwise vorticity convection, whereas the spanwise gradient of spanwise flow dictates the vortex stretching. Some research reported that swept wings can enhance both spanwise flow and vorticity convection (Onoue & Breuer Reference Onoue and Breuer2017; Adhikari et al. Reference Adhikari, Loubimov, Kinzel and Bhattacharya2022), thus avoiding excessive LEV growth. However, other research believed the enhanced spanwise flow can promote both vorticity convection and vortex stretching (Wong & Rival Reference Wong and Rival2015; Zangeneh Reference Zangeneh2021). This remains an unsolved issue for explaining the benefits of a nonlinear leading-edge profile in flapping wings and fins.
\begin{equation} \partial {\boldsymbol{\omega }}/\partial {\tau }=-(\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{\nabla })\boldsymbol{\omega }+(\boldsymbol{\omega }\boldsymbol{\cdot }\boldsymbol{\nabla })\boldsymbol{u}+{\nabla} ^2\boldsymbol{\omega }/Re. \end{equation}
Reviewing the conclusion of Wong et al. (Reference Wong, Kriegseis and Rival2013), the condition that an enhanced spanwise flow can promote LEV stability, i.e. a spanwise gradient of effective incidence, contains two intertwined factors. The effective angle of attack and incident velocity both experience a spanwise gradient in their work, which makes it hard to clarify the more critical factor. Aiming at addressing the fundamental physics of swept flapping wings and fins, the current work focuses on the case of partially swept wings undergoing revolving motion, which retains the combination of a swept geometry and a spanwise gradient of incident velocity, while the effective angle of attack is fixed along the span. The swept wings in this research are simplified by rectangular plates with outboard edges subject to either linear or nonlinear swept profiles.
The research framework is summarised here. In § 2, the simplified geometric model, revolving kinematics, numerical method and data post-processing method are introduced. The LEV dynamics is discussed in § 3.1, including three-dimensional (3-D) structures, near-wall flow patterns, LEV circulations and vortex movements at selected spanwise locations. The enhancement of spanwise flow is then quantified in § 3.2, followed by a spatio-temporal analysis of vorticity convection and vortex stretching (§ 3.3). The link between the LEV dynamics, pressure distributions and aerodynamic forces is discussed in § 3.4. Finally, the conclusions are addressed in § 4.
2. Methodology
2.1. Swept wing models
To reproduce the swept feature of natural flapping wings and fins, a rectangular plate with a swept region covering the outboard 50
$\%$
span is employed here. According to Lentink & Dickinson (Reference Lentink and Dickinson2009a
), the AR (wing span b over chord c) of most natural flapping wings is around 3, thus the AR of the plate is 3. As shown in figure 3(a), both linear and nonlinear swept wings are considered. The profiles of leading and trailing edges are identical, and the span is fixed to eliminate a concomitant change of wing area. This also ensures a constant chord along the span. The profile of a linear leading edge is determined by the sweep angle (
$\beta$
), which varies from 0
$^{\circ }$
(straight) to 40
$^{\circ }$
to cover most natural wings (Zurman-Nasution et al. Reference Zurman-Nasution, Ganapathisubramani and Weymouth2021). For nonlinear profiles, the leading-edge deflection follows a quadratic rule and is determined by the terminal deflection at the wing tip (
$\Delta$
c). The local deflection of the leading edge with respect to the baseline (
$c_{\textit {le}}$
(z)) is thus calculated as
\begin{equation} c_{le}(z)= 4 \Delta c (z-0.5b-z_{0})^{2} / b^{2}. \end{equation}
To compare with linear profiles, four
$\Delta$
c values are selected corresponding to the wing-tip deflection of the linear profiles with
$\beta$
= 0
$^{\circ }$
–40
$^{\circ }$
. This indicates that the leading-edge slopes at the 75
$\%$
span of nonlinear profiles are equal to those of the linear profiles. For both sorts of swept wings, the thickness is 0.03c, and a root cutoff (z
$_{0}$
) of 0.9c is included to model the connection between the wing and the body of natural flyers.
Geometry and kinematics of swept wing models: (a) linear and nonlinear leading-edge profiles, (b) coordinate definition and (c) revolving motion.
The coordinates and wing motion are shown in figures 3(b) and 3(c). The wing-fixed frame is defined by base axes (
$\boldsymbol{e_x}$
,
$\boldsymbol{e_y}$
,
$\boldsymbol{e_z}$
), and the inboard spanwise direction is aligned with
$\boldsymbol{e_z}$
. To retain this alignment, the local frame in the outboard region (
$\boldsymbol{e_x^*}$
,
$\boldsymbol{e_y^*}$
,
$\boldsymbol{e_z^*}$
) is rotated around
$\boldsymbol{e_y}$
, and the rotation angle equals the local slope of the leading edge. The vectors in the outboard region can thus be re-projected into the local Cartesian frame using a rotation matrix. In this research, all wings are subject to a constant-speed revolving motion with a fixed angle of attack (
$\alpha$
) of 45
$^{\circ }$
. A dimensionless time scale
$\lambda$
, defined as the chord length travelled by the wing section at the radius of gyration (
$R_g$
), is introduced to describe the formation of LEV, as follows:
\begin{equation} \lambda =\varphi R_g/c ,\end{equation}
Numerical methods: (a) the O-H computational mesh and (b) spanwise slices, LEV region and integrations. Two slices at 25
$\%$
and 75
$\%$
span are shown here as representatives.
and
\begin{equation} R_g=\sqrt {\int _{z_0}^{b+z_0}z^2\mathrm{d}z/b}. \end{equation}
Here,
$\phi$
is the revolving angle of the wing. An abrupt ramp is employed to speed up the wing to a constant speed. The Reynolds number (
$Re$
) is defined by the constant speed (
$\varOmega _0$
) at
$R_g$
,
$Re=\varOmega _0 R_g c/\nu$
, in which
$\nu$
is the kinematic viscosity. This revolving motion triggers a prominent spanwise gradient of incident velocity at a fixed
$\alpha$
. Note that the constant-speed revolving motion may ignore some fluid physics of flapping wings, especially during the pronation and supination (concomitant with the start/stop periods). However, it can retain the physics related to the revolving kinematics, which is one of the most important factors that stabilise the LEV. Similar revolving models have also been employed in previous research relevant to flapping wings and fins (Lentink & Dickinson Reference Lentink and Dickinson2009b
; Garmann et al. Reference Garmann, Visbal and Orkwis2013; Harbig et al. Reference Harbig, Sheridan and Thompson2013; Jones et al. Reference Jones, Medina, Spooner and Mulleners2016; Jardin Reference Jardin2017; Chen et al. Reference Chen, Wang, Zhou, Wu and Cheng2022, Reference Chen, Zhou, Werner, Cheng and Wu2023c
).
2.2. Numerical simulation
The flow generated by revolving wings is simulated by an in-house solver that has been validated in our previous works (Chen et al. Reference Chen, Wu and Cheng2020a , Reference Chen, Wang, Zhou, Wu and Cheng2022, Reference Chen, Zhou, Werner, Cheng and Wu2023c ). The governing equations are the 3-D incompressible unsteady Navier–Stokes equations
\begin{equation} \left. \begin{array}{ll} \displaystyle \boldsymbol{\nabla }\boldsymbol{\cdot }{\boldsymbol{u}}=0\\ \displaystyle \partial {\boldsymbol{u}}/\partial {\tau }+(\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{\nabla })\boldsymbol{u}+\boldsymbol{\nabla }{p}-{\nabla} ^2\boldsymbol{u}/Re=\boldsymbol{0} \end{array}\right \}, \end{equation}
where
$\boldsymbol{u}$
and p are dimensionless velocity and pressure.
The solver allocates primary variables in a node-collocated fashion and ensures second-order accuracy in both temporal and spatial domains. A cylindrical flow domain is defined with its axis parallel to the wing span, and a typical O-H mesh is generated to discretise the domain (figure 4
a). The domain size is 25c in the radial and spanwise directions. The mesh density has been well validated, and the refined mesh contains 81 × 81 × 91 nodes in wing-normal, chordwise and spanwise directions. The boundary layer set-up and time interval have been validated to guarantee a converged solution and to identify the transient near-wall flows. The height of the first layer from the surface is 0.001c, and a
$360^{\circ }$
revolving cycle is split into 1600 time steps. According to our experiments of revolving wings (Wu et al. Reference Wu, Chen, Zhou, Hsu and Cheng2019) and pitching–flapping perturbed revolving wings (Chen et al. Reference Chen, Wu, Zhou, Hsu and Cheng2018, Reference Chen, Zhou and Wu2020b
, Reference Chen, Cheng, Zhou, Zhang and Wu2024), the numerical method can provide a good resolution of both flow properties and aerodynamic force.
2.3. Data analysis
The LEV evolution is underpinned by the 3-D vorticity transport inside and near the vortex. To better illustrate the revolving nature of flow patterns, the vorticity transport equation in the inertial frame (1.1) is transformed into the wing-fixed co-rotating frame
\begin{equation} \partial {\boldsymbol{\omega }^*}/\partial {\tau }=-(\boldsymbol{u}^*\boldsymbol{\cdot }\boldsymbol{\nabla })\boldsymbol{\omega }^*+(\boldsymbol{\omega }\boldsymbol{\cdot }\boldsymbol{\nabla })\boldsymbol{u}^*+{\nabla} ^2\boldsymbol{\omega }^*/Re, \end{equation}
where
$\boldsymbol{\omega }^*$
is the vorticity in the wing-fixed frame and equals
$\boldsymbol{\omega }-2\boldsymbol{\varOmega }$
(
$\boldsymbol{\omega }$
and
$\boldsymbol{\varOmega }$
are the total vorticity and angular rotation speed). Here,
$\boldsymbol{u}^*$
is the relative velocity and equals
$\boldsymbol{u}-\boldsymbol{\varOmega }\times \boldsymbol{r}$
(
$\boldsymbol{r}$
is the position vector). The vorticity convection, vortex tilting and stretching and vorticity diffusion are defined subsequently by the terms on the right-hand side of (2.5). Based on our analysis (Chen et al. Reference Chen, Wang, Zhou, Wu and Cheng2022), the evolution of the spanwise vorticity component mostly represents the LEV dynamics. Thus, this work focuses on the spanwise component of vorticity transport. Equation (2.5) is then projected into the spanwise direction of the local Cartesian frames (
$\boldsymbol{e_z}$
or
$\boldsymbol{e_z^*}$
)
\begin{equation} \partial {\omega _z^*}/\partial {\tau }=-(\boldsymbol{u}^*\boldsymbol{\cdot }\boldsymbol{\nabla })\omega _z^*+(\boldsymbol{\omega }\boldsymbol{\cdot }\boldsymbol{\nabla })u_z^*+{\nabla} ^2\omega _z^*/Re. \end{equation}
As one of the motivations of our research, the vorticity convection and vortex stretching in the spanwise direction are examined and compared in detail. According to the definition of local Cartesian frames, the spanwise vorticity transport on a swept wing can be quantified by
$-u_z^*\partial {\omega _z^*}/\partial {z}$
(vorticity convection, C) and
$\omega _z^*\partial {u_z^*}/\partial {z}$
(vortex stretching, S). Thus, the spatio-temporal analysis of these two terms is presented, whereas other terms are ignored.
To illustrate the spanwise variation of the LEV dynamics and vorticity transport, a series of spanwise slices cut through the flow domain (figure 4
b). Note that the spanwise slices beyond the mid-span are perpendicular to the local mid-chord lines. The vortex region on each slice can be outlined using the vortex identification index Q, i.e. the second invariant of velocity gradient (Jeong & Hussain Reference Jeong and Hussain1995). According to a sensitivity study, the Q threshold is determined at 1, and a secondary constraint (
$\omega _z^*\lt$
0) is included to separate the LEV region
$\varSigma _1$
. In the following analysis, the strong spanwise flow may not be limited within the LEV region
$\varSigma _1$
but cover the entire dorsal side. Therefore, another rectangular integration boundary
$\varSigma _2$
is introduced. This boundary ranges from the leading edge to 1c aft of the trailing edge and covers 0.02c–0.5c above the dorsal surface. The integrals calculated within selected boundaries are labelled by brackets, as shown in figure 4(b).
Although the Q threshold can outline the vortex boundary, it fails to locate the vortex core. Therefore, the robust algorithm developed by Graftieaux, Michard & Grosjean (Reference Graftieaux, Michard and Grosjean2001) is employed to resolve the vortex core within each cylindrical slice. In their algorithm, a 5 × 5 (total points N = 25) window scans over each slice to calculate a scalar field
$\gamma _1$
from the velocity field (2.7)
\begin{equation} \gamma _1(q)=\frac {1}{N}\sum _{i=1}^{N}{[(((\boldsymbol{x}_i-\boldsymbol{x}_q)\times \boldsymbol{u}_i^*)\boldsymbol{\cdot }\boldsymbol{n})/(\|\boldsymbol{x}_i-\boldsymbol{x}_q\|\boldsymbol{\cdot }\|\boldsymbol{u}_i^*\|)]}. \end{equation}
Here, q denotes the centre of the window, and i represents an arbitrary point within the window. Also,
$\boldsymbol{x}$
is the position vector on the slice, and
$\boldsymbol{n}$
is the unit vector in the normal direction of the slice. The LEV core is thus located at the
$\gamma _1$
maximum position where
$\omega _z^*$
is below 0.
3. Results and discussion
Our discussion about the impacts of swept leading edges on flow features is threefold: (i) characterise the LEV structure, intensity and location, (ii) quantify the spanwise flow enhancement and (iii) elucidate changes in the spatio-temporal evolution of spanwise vorticity convection and vortex stretching. The lift generation and underlying pressure distribution are further discussed to connect with flow features.
3.1. Leading-edge vortex dynamics
3.1.1. Three-dimensional vortical structures
Three snapshots during the evolution of 3-D vortical structures on all wings are compared in figure 5. According to the LEV formation process at
$Re$
= 1500 (Chen et al. Reference Chen, Wang, Zhou, Wu and Cheng2022), these snapshots represent the LEV growth (
$\lambda$
= 2), the arrival of the steady state (
$\lambda$
= 4) and post-steady-state fluctuation (
$\lambda$
= 6), respectively. For a rectangular wing (figure 5
a), the LEV in the outboard region experiences remarkable intermittent shedding, and, unlike the inboard stable LEV, leads to an unsteady LEV region (see red dashed lines in figure 5
a). Once the outboard region of the wing is swept backward under either linear or nonlinear profiles, the stable LEV structure and its vorticity components are barely changed (figures 5
b and 5
c). The swept leading edges mostly manipulate the vortical structures beyond the mid-span.
Evolution of vortical structures on: (a) rectangular, (b) linear swept and (c) nonlinear swept wings. Three snapshots are labelled with coloured boxes. The RGB of the isosurface correspond to the magnitudes of vorticity components:
$\omega _x^*$
in red,
$\omega _y^*$
in green and
$\omega _z^*$
in blue.
One of the major features attributed to a swept leading edge is the connection of the LEV and the tip vortex. Previous research illustrated that the LEV on a rectangular wing was pinned to the dorsal surface near the corner of the wing tip (Garmann & Visbal Reference Garmann and Visbal2014), leading to a low-pressure footprint. The vortex lines of the LEV and tip vortex are thus disconnected. However, a swept leading edge prevents a pinned LEV and promotes the connection of the vortex line with the tip vortex (figures 5
b and 5
c,
$\lambda$
= 2, red boxes). This is in line with the observations on plunging swept wings in a free stream (Beem et al. Reference Beem, Rival and Triantafyllou2012). Another notable change in the outboard vortical structure is a spanwise elongation in shape, replacing the conventional conical structure with a more cylinder-like structure. However, the vorticity components in the outboard vortical structure are mostly retained and are dominated by a mixture of spanwise (negative) and chordwise vorticity components.
As the LEV approaches the steady state (
$\lambda$
= 4, blue boxes), the unsteady LEV shedding still occurs in the outboard region of a swept wing, while the complex twine of LEV and tip vortex is attenuated. The LEV shed near the tip of a swept wing takes shape as a single vortical structure consisting of spanwise and chordwise vorticity components. As the LEV and tip vortex are connected in their growth, less mutual vortex interaction is involved in a swept wing. When comparing a linear profile with a nonlinear one (at the same final deflection), the LEV on a linear one is further elongated and compact before its shedding.
For the post-steady-state evolution (
$\lambda$
= 6, yellow boxes), a swept leading edge can enhance the chordwise vorticity component in the outboard vortical structure, sustaining a comparable strength of spanwise and chordwise vorticity components. This phenomenon is more remarkable for a linear swept wing with a sufficiently large sweep angle (
$\beta$
= 40
$^{\circ }$
). According to our previous findings (Chen et al. Reference Chen, Zhou, Werner, Cheng and Wu2023c
), the balanced vorticity in spanwise and chordwise directions implies the existence of a dual-stage radial–tangential vortex tilting mechanism that can regulate the local growth of spanwise vorticity and thus contributes to local LEV stability. This may explain the limited expansion of vortical structures in the outboard region of swept wings.
For both linear and nonlinear swept wings, an increase in the sweep angle
$\beta$
(or the wing-tip terminal deflection
$\Delta c$
) only leads to quantitative differences in 3-D vortical structures. Thus, the swept wings with the highest
$\beta$
and
$\Delta c$
are chosen as representatives for two swept profiles in the following analysis.
3.1.2. Near-wall flow patterns
To zoom into the flow features inside the vortical structures, the streamlines on a plane 0.007c above the dorsal surface are shown in figure 6, together with the contours of spanwise velocity
$u_z^*$
. According to the critical-point concepts for flow topology analysis (Perry & Chong Reference Perry and Chong1987; Phillips et al. Reference Phillips, Knowles and Bomphrey2015), two of the most common separation patterns, i.e. the open negative bifurcation line separation and the Werlé–Legendre separation, are mostly involved in natural flapping wings. The first pattern refers to a flow convergence to a separatrix, while the latter one manifests as spiral flows into a focus plus sink, indicating that it enters the surface as a vortex axis originating from the focus–sink centre. Together with the negative bifurcation line, a reattachment line is also shown here to outline the LEV footprint.
Flow topology on the dorsal surface of: (a) rectangular, (b) linear swept (
$\beta$
= 40
$^{\circ }$
) and (c) nonlinear swept (
$\Delta c$
= 1.26) wings. TEV-trailing-edge vortex.
For the rectangular wing (figure 6
a), the LEV, limited within the negative and positive bifurcation lines (white and red lines), is almost attached within the entire span at
$\lambda$
= 1. A clear correlation between the LEV and spanwise flow is also demonstrated. As the LEV approaches the steady state (
$\lambda$
= 2 and 4), the reattachment line is shifted downstream and first reaches the trailing edge in the outboard region. This implies a normal stall with the formation of a trailing-edge vortex, which is regarded as the trigger for LEV shedding (Harbig et al. Reference Harbig, Sheridan and Thompson2013; Wong & Rival Reference Wong and Rival2015). Again, the stable LEV and strong spanwise flow are both confined within the mid-span. Moreover, a Werlé–Legendre separation (foci–sink centre) occurs in the outboard region at
$\lambda$
= 4, indicating a strong lift-off of the LEV and a vortex axis normal to the dorsal surface. However, due to the strong vortex interaction in the outboard region, the foci–sink centre becomes absent at
$\lambda$
= 6.
For a linear swept wing (figure 6
b), the negative bifurcation line indicating the separation becomes aligned with the leading-edge profile and extends towards the wing tip. In contrast, the reattachment line is terminated at the mid-span where the leading edge starts to sweep back (
$\lambda$
= 1–2), and, more interestingly, the stagnation points aft of the LEV (shown by the reattachment line) are locked on the trailing edge beyond the mid-span. This implies a faster growth of a full-size LEV within this region. The spanwise flow in the outboard region is enhanced by the swept leading edge and reaches a strength comparable to that in the inboard region (
$\lambda$
= 2). As the LEV grows (
$\lambda$
= 4–6), the full-size LEV is first retained, and then the stall occurs starting from the trailing edge. The spanwise flow is also compressed towards the inboard region, and a stronger reversed chordwise flow is observed near the trailing edge. This coincides with a closely attached TEV at the corner where the trailing edge starts to sweep back.
The formation and sustained attachment of this TEV, which does not occur for a straight trailing edge, can originate from the alignment of the spanwise flow with the slope of the trailing edge. For a rectangular wing, the spanwise flow is slowed down by the adverse pressure gradient induced by the tip vortex. However, for a linear swept wing, the tip vortex is reduced in strength (figure 5 b) and thus the spanwise flow is less decelerated by the adverse pressure gradient. Thus, once the spanwise flow reaches the trailing edge, the residual energy is still sufficient to transport vorticity (which is positive for the TEV) away, and its direction is almost parallel to the TEV axis. This vorticity convection helps to stabilise the local TEV in a manner similar to the LEV and thereby avoids excessive roll-up.
For a nonlinear swept wing (figure 6 c), no qualitative difference is observed in the evolution of near-wall flow patterns, except that, compared with the linear swept wing, the attached TEV is smaller in size and weaker in strength. The arch-shaped trailing edge is less aligned with the spanwise flow and may also lead to novel vortex stretching effects, thus manipulating the TEV evolution. One of the advantages caused by this attached TEV is an additional suction force that enhances the overall lift generation. This will be discussed in § 3.4.
3.1.3. LEV circulation and location
To quantify the LEV strength and its spatial movement, the evolution of circulation (
$\varGamma$
) at 50
$\%$
and 75
$\%$
span is shown in figure 7, together with the height of the LEV core above the dorsal surface (
$l_s$
). The value of
$\varGamma$
is estimated by an integral of spanwise vorticity inside the
$\varSigma _1$
boundary, which is outlined by the black lines in figures 7(e) and 7(f).
Spatio-temporal evolution of the LEV on representative slices: (a) and (b) LEV intensity, (c) and (d) height of the LEV core above the wing, (e) and (f) vorticity contour. Left column: 50
$\%$
span, right column: 75
$\%$
span.
According to figure 7(a), a swept leading edge can reduce the steady-state circulation at 50
$\%$
span, and a linear swept profile with
$\beta$
= 40
$^{\circ }$
shows the strongest reduction up to 25
$\%$
. Moreover, this regulation on LEV intensity starts during its growth period and barely affects the critical chord length of travel to reach the steady state (Chen et al. Reference Chen, Wu and Cheng2019). In contrast, the circulation at 75
$\%$
span is enhanced by both linear and nonlinear swept profiles during the LEV formation (
$\lambda$
= 1–2, figure 7
b). For a rectangular wing, the
$\varGamma$
at 75
$\%$
span continuously increases during
$\lambda$
= 4–6 due to the formation of a secondary LEV. However, the corresponding
$\varGamma$
of both linear and nonlinear swept wings starts to decrease. These findings quantitatively prove that sweeping back the outboard leading edge of a revolving wing can limit excessive growth of inboard LEV circulation by removing vorticity into the outboard region. This can be attributed to the stronger spanwise convection led by an enhanced spanwise flow. Apart from limited circulation, swept leading edges can also push the stable LEV further towards the dorsal surface. This is supported by the tighter attachment at 50
$\%$
span and, again, the linear profile outperforms the nonlinear one (figure 7
c). More importantly, the quick lift-off of the local LEV at 75
$\%$
span of the rectangular wing is slowed down (
$\lambda$
= 0–2, figure 7
d). Thus, the primary LEV shedding is postponed and no remarkable LEV shedding occurs for the linear swept wing (movement of the secondary LEV is labelled by red lines in figure 7
d).
The steady-state vorticity contours at both sections are then compared to show the impact of swept leading edges on LEV evolution. At 50
$\%$
span (figure 7
e), the LEV remains attached to the dorsal surface in all cases. However, the local LEV on the rectangular wing is slightly diffused while those on the swept wings retain a compact structure with vorticity concentrated in the vortex core (
$\lambda$
= 6). Moreover, this compact LEV is closely attached to the dorsal surface. At 75
$\%$
span (figure 7
f), the local primary LEV on the rectangular wing is detached at
$\lambda$
= 4, and then the secondary LEV occupies the space behind the wing at
$\lambda$
= 6. This unsteady flow feature is remarkably limited on swept wings, in that only a small portion of the primary LEV sheds into the wake, and the residual vorticity merges with the secondary LEV. In general, this results in a slight decrease of circulation in the outboard region, relieving the sudden drop in local
$l_s$
(figure 7
d). Although the outboard LEV becomes more stable and closer to the wing, it may not increase lift generation since the local
$\varGamma$
is reduced.
Another remarkable flow feature on the swept wings is the formation of an attached TEV in the outboard region. As shown in figure 7
f, the trailing-edge shear layer rolls up into a TEV on the rectangular wing at
$\lambda$
= 4, and it is quickly shed into the wake at
$\lambda$
= 6. However, the TEVs of both swept wings stay close to the trailing edge once they are generated (
$\lambda$
= 4–6).
3.2. Spanwise flow enhancement
A particular benefit of swept wings that may contribute to LEV stability is an enhanced spanwise flow due to the decomposition of incoming velocity (figure 2). This section elucidates the spanwise flow enhancement and its dependency on swept profiles. The spatio-temporal evolution of spanwise flow on three typical cases is compared in figure 8. In general, the spanwise flow region in the downstream expands into the wake for both swept wings (figure 8
a). Due to the prominent LEV shedding at
$\lambda$
= 4–6, the spanwise flow at the 75
$\%$
span of the rectangular wing is limited to a thin layer above the dorsal surface. However, the spanwise flow at the identical section of swept wings is significantly stronger during
$\lambda$
= 2–4, and it occupies a larger wake region extending up to 1c away from the trailing edge. At
$\lambda$
= 2, a good matching of the strong spanwise flow and the LEV region is observed in the 75
$\%$
span of all cases. However, as the wing rotates further, the local LEV tends to shed into the wake, and thus it is less overlapped with the strong spanwise flow region at
$\lambda$
= 4 and 6. Compared with the nonlinear swept wing, the linear swept wing can produce a stronger spanwise flow, but the spatial distributions of spanwise flow are close for both cases. This is because the leading-edge slope within the 50
$\%$
–75
$\%$
span of the linear swept wing (
$\beta$
= 40
$^{\circ }$
) is steeper, thus increasing the spanwise projection of incoming velocity.
Spatio-temporal evolution of spanwise flow: (a) iso-surfaces and contours at 75
$\%$
span, (b) intensity inside the
$\varSigma _2$
boundary at 75
$\%$
span and (c) intensity inside the
$\varSigma _1$
boundary at 75
$\%$
span. The iso-surfaces in panel (a) are drawn by
$u_z^*$
= –1.
A quantitative assessment of spanwise flow intensity is then shown in figures 8(b) and 8(c). As shown in figure 8(a), the spanwise flow is less overlapped with the local LEV region at
$\lambda$
= 4 and 6, thus an integration within the
$\varSigma _1$
boundary may not consider the bulk of the spanwise flow. To mostly enclose the spanwise flow region at 75
$\%$
span, the
$\varSigma _2$
boundary is employed for the integration, and
$[u_z^*]$
considers both inboard and outboard flows. Note that an integration of spanwise flow within the
$\varSigma _1$
boundary is also shown in figure 8(c) to relate with the integration of vorticity and vorticity transport.
As shown in figure 8(b), the enhancement of spanwise flow is remarkable since the LEV formation (
$\lambda$
= 1), which can be up to 150
$\%$
in strength for the linear swept wing (
$\lambda$
= 2). This enhancement is retained as the LEV enters the steady state (
$\lambda$
= 4–6). Compared with the nonlinear swept wing, the linear swept wing can lead to a stronger spanwise flow given the same wing-tip deflection. The spanwise flow enhancement on swept wings is also shown by the integration within the LEV region (
$\lambda \lt$
2, figure 8
c), and the linear swept wing still outperforms the nonlinear swept wing. However, due to the shedding of local LEV, the enhanced spanwise flow near the dorsal wing surface is less captured by the LEV region (
$\lambda$
= 4–6, figure 8
a). Therefore, the local spanwise flow of all three wings tends to decrease since
$\lambda$
= 2 and experiences strong unsteady variations until
$\lambda$
= 6.
Effects of swept slope (2
$\Delta$
c/AR) on the spanwise flow enhancement: (a) 62.5
$\%$
span, (b) 75
$\%$
span and (c) 87.5
$\%$
span. Here,
$\Delta [u_z^*]$
is the change of spanwise flow intensity compared with the rectangular wing
$[u_z^*]_{{rec}}$
and
$\Delta [u_z^*]$
and
$[u_z^*]_{{rec}}$
are calculated inside the
$\varSigma _2$
boundary.
To demonstrate the impacts of leading-edge slope on the formation of spanwise flow, the relative variation of spanwise flow intensity (with respect to the rectangular wing) at three representative spanwise locations are displayed over the swept slopes (figure 9). Here, the swept slope is defined as the ratio of the final deviation at the wing tip over the half-span, representing the slope of a virtual line that connects the mid-span point with the wing-tip corner (both at the leading edge). As shown in figure 9(a), a swept wing with either linear or nonlinear profile experiences slight spanwise flow enhancement at 62.5
$\%$
span (
$\lambda$
= 4). At further outboard locations, e.g. 75
$\%$
and 87.5
$\%$
span (figures 9
b and 9
c), the spanwise flow enhancement becomes prominent. This increment can be up to over 100
$\%$
during the early period of the LEV formation (
$\lambda$
= 1) and tends to be diminished as the LEV approaches the steady state (
$\lambda$
= 3–4). Note that, although the overall intensity of the spanwise flow in the wake region is close for all cases, their spatial distributions can be different (figure 8
a,
$\lambda$
= 4 and 6). Moreover, the spanwise flow enhancement led by a swept profile (either linear or nonlinear) shows an approximately linear dependency on the swept slope. Linear profiles outperform nonlinear ones at 75
$\%$
span, while they become comparable at 87.5
$\%$
span. This is because the local slope of a nonlinear profile keeps increasing over the span, enlarging the projection of incoming velocity in the spanwise direction.
3.3. Vorticity transport analysis
3.3.1. Spanwise convection
Swept leading edges can induce a stronger spanwise flow in the wake, but their impacts on the critical vorticity transport remain unclear. This is because an enhanced spanwise flow is not sufficient to infer a stronger vorticity convection or vortex stretching in the spanwise direction (Wong & Rival Reference Wong and Rival2015; Chen et al. Reference Chen, Wang, Zhou, Wu and Cheng2022). Here, a spatio-temporal analysis of spanwise vorticity convection is first uncovered. The evolution of spanwise convection among rectangular and swept wings is compared in figure 10. For the rectangular wing (figure 10
a), no obvious spanwise convection exists during the early stage of rotation (
$\lambda$
= 1–2), except for the wing-tip region, in which the unsteady tip vortex formation leads to a strong positive [C] . As the LEV grows, two regions with different [C] behaviours are observed within the steady LEV region, i.e. a balanced [C] (region I) near the wing root and a negative [C] (region II) around the mid-span. This implies the existence of spanwise vorticity convection within the LEV (Chen et al. Reference Chen, Wang, Zhou, Wu and Cheng2022), and the spanwise vorticity is continuously transported from region I to region II. Thus, the spanwise convection C within region I is almost balanced. Note that the negative [C] in region II implies that the spanwise vorticity from region I is locally accumulated and should be balanced by other vorticity transport effects, such as vortex tilting.
Spatio-temporal analysis of spanwise convection: (a) rectangular, (b) linear swept (
$\beta$
= 40
$^{\circ }$
) and (c) nonlinear swept (
$\Delta c$
= 1.26) wings. Here,
$[C]$
is calculated inside the
$\varSigma _1$
boundary.
For the linear swept wing with
$\beta$
= 40
$^{\circ }$
(figure 10
b), one of the most remarkable benefits is that the negative [C] in region II is significantly reduced and thus a balanced [C] is achieved within a larger spanwise region. This implies that the removal of spanwise vorticity, which mostly occurs in region I, further expands beyond the mid-span and can transport local vorticity towards the wing tip. This thus proves the enhanced spanwise convection on the linear swept wing. In comparison, for the representative nonlinear swept wing (figure 10
c), the balanced [C] in region I is also expanded towards the wing tip, and the negative [C] in region II is thus weakened. However, the enhancement of spanwise convection led by the nonlinear leading-edge profile is inferior to that from the linear leading-edge profile.
Comparison of spanwise convection sub-terms: (a) contours at 50
$\%$
span, (b) contours at 75
$\%$
span and (c) spatio-temporal analysis of positive and negative sub-terms.
To further elucidate the evolution of spanwise convection in these cases, the spanwise snapshots of C at 50
$\%$
and 75
$\%$
span are shown in figures 11(a) and 11(b). As explained in our previous research (Chen et al. Reference Chen, Wang, Zhou, Wu and Cheng2022), the spanwise vorticity convection (for LEV, shown by the convection from positive C to negative C) at the mid-span of a rectangular wing becomes attenuated as the LEV approaches the steady state (
$\lambda$
= 6, figure 11
a). During the LEV formation process (
$\lambda$
= 1–4), the positive C occurs near the downstream boundary of the LEV. In contrast, for both linear and nonlinear swept wings, the transient positive C within
$\lambda$
= 1–4 is intensified and covers a larger chordwise range. More importantly, this positive C is sustained when the LEV reaches the steady state (
$\lambda$
= 6), indicating an enhanced spanwise vorticity convection that improves local LEV stability. This enhancement manifests as an extended region of the balance spanwise convection along the span (figure 10
b).
The spanwise extension of the vorticity convection is proved by the snapshots at 75
$\%$
span (figure 11
b). For a rectangular wing, almost no positive C forms near the LEV during the entire evolution. In contrast, the positive C near the downstream boundary of the LEV is retained for both swept wings at
$\lambda$
= 1–2 and is then surrounded by a negative C near the trailing edge (
$\lambda$
= 4–6). Since an attached TEV (with positive spanwise vorticity) is observed near the trailing edge of swept wings (figure 7
f), the negative C indicates the removal of vorticity inside the TEV and contributes to TEV attachment. Thus, it is concluded that the enhanced spanwise flow on swept wings can intensify the vorticity convection in the spanwise direction, leading to both reduced LEV strength (figure 7
a) and a sustained attached TEV beyond the mid-span.
A decomposed quantitative analysis of positive and negative spanwise convection is conducted in figure 11(c). Here,
$[C^{(+)}]$
and
$[C^{(-)}]$
denote the integrals of positive and negative C inside the
$\varSigma _1$
boundary. For a rectangular wing, the spanwise misalignment between the strong
$[C^{(+)}]$
and
$[C^{(-)}]$
clearly shows the vorticity convection that benefits LEV stability. This stabilising mechanism occurs starting from
$\lambda$
= 2 and settles within the mid-span at the steady state. In contrast, a linear swept wing can enlarge both strong
$[C^{(+)}]$
and
$[C^{(-)}]$
regions towards the wing tip. The vorticity convection within the mid-span of a linear swept wing is similar to that of a rectangular wing, while novel peaks of both
$[C^{(+)}]$
and
$[C^{(-)}]$
occur beyond the mid-span (regions I.1 and II.1). Moreover, remarkable
$[C^{(+)}]$
and
$[C^{(-)}]$
occurs in the swept region (regions I.2 and II.2) since the onset of the revolving motion. Note that the vorticity convection in the spanwise direction of a rectangular wing is established after a certain revolving distance since a sufficiently strong spanwise flow should be first generated. However, the swept wing can directly produce a spanwise flow via the velocity decomposition, giving birth to the vorticity convection in the outboard region at the beginning of motion. This is also consistent with the locally balanced [C] shown in figure 10(b).
Compared with the linear swept wing, the nonlinear swept wing experiences a similar change in the sub-terms of spanwise convection (figure 11
c). Although the improvement of
$[C^{(+)}]$
is less significant than that of the linear swept wing, it still outperforms the rectangular wing and leads to the formation of both regions I.1 and I.2. Since the swept slope near the mid-span of the nonlinear swept wing is lower than the linear one, the local
$[C^{(+)}]$
peaks are not remarkable. Also, the occurrence of the
$[C^{(+)}]$
region at the start of the motion is closer to the wing tip, where a higher local swept slope is imposed.
3.3.2. Vortex stretching
The impacts of swept profiles on the spanwise vortex stretching (S) are discussed in this section. As demonstrated in our previous work (Chen et al. Reference Chen, Wang, Zhou, Wu and Cheng2022), the spanwise vortex stretching is linked to the spanwise gradient of spanwise flow, instead of its magnitude. thus, an enhanced spanwise flow may not lead to a preferred variation in vortex stretching. The integral boundary in this section is identical to that in § 3.3.1, and the evolution of spanwise vortex tilting among three representative wings is compared in figure 12. In general, the LEV in the inboard region of the rectangular wing is enhanced by vortex stretching (for LEV, negative [S] denotes vortex stretching), as shown in figure 12(a). Similar to the spanwise convection, this enhancement is obvious since
$\lambda$
= 2. In contrast, the LEV near the mid-span is limited by vortex compression (positive [S]), which is caused by the adverse pressure gradient originating from the wing tip (Chen et al. Reference Chen, Wu and Cheng2019, Reference Chen, Wang, Zhou, Wu and Cheng2022). Note that the LEV contains negative spanwise vorticity and thus a positive S indicates vortex compression. According to the explanation of Wong & Rival (Reference Wong and Rival2015), vortex compression is detrimental to LEV attachment since it expands the vortex size, and thus the LEV approaches the critical length scale c to trigger vortex shedding. For both linear and nonlinear swept wings (figures 12
b and 12
c), the vortex compression inside the LEV (region II) is attenuated near the mid-span, despite the fact that more remarkable vortex compression occurs in the outboard region. Moreover, the vortex stretching in region I is further enhanced and elongated in the spanwise direction. This implies that the LEV can be further stretched in the inboard region and near the mid-span, and thus indicates a profit for its prolonged attachment along the span.
Spatio-temporal analysis of vortex stretching (compression) in the spanwise direction: (a) rectangular, (b) linear swept (
$\beta$
= 40
$^{\circ }$
) and (c) nonlinear swept (
$\Delta c$
= 1.26) wings. Here,
$[S]$
is calculated inside the
$\varSigma _1$
boundary.
Comparison of spanwise vortex stretching (compression) sub-terms: (a) contours at 50
$\%$
span, (b) contours at 75
$\%$
span and (c) spatio-temporal analysis of positive and negative sub-terms.
The spanwise snapshots of S at 50
$\%$
and 75
$\%$
span are compared in figures 13(a) and 13(b). An obvious vortex compression appears inside the LEV at the mid-span of the straight wing and becomes stabilised as the LEV reaches the steady state (figure 13
a,
$\lambda$
= 4–6). In contrast, the vortex compression inside the LEV is attenuated at the mid-span of swept wings. Alternatively, a vortex stretching is established near the downstream boundary of the LEV and is sustained until the steady state (
$\lambda$
= 4–6). This stretching effect helps to elongate the vortex along its line and thus shrinks its local area, again proving that a sufficiently swept wing can produce vortex stretching that limits an excessive expansion of vortex size towards the trailing edge. Moreover, this vortex stretching is more prominent for a linear swept wing given a prescribed final deflection at the wing tip. This is because the vortex stretching at the mid-span of the nonlinear swept wing with
$\Delta c$
= 1.26 is almost absent, while that on the linear swept wing with
$\beta$
= 40
$^{\circ }$
is retained. At the 75
$\%$
span of the rectangular wing (figure 13
b), the vortex compression is quickly diffused as the LEV forms (
$\lambda$
= 1–2) and its following contribution within the LEV region becomes fully unsteady. However, this diffusion of vortex compression can be delayed on both swept wings since a compact region of positive S still exists within the local LEV at
$\lambda$
= 2. This also implies that a swept wing can move the strong vortex compression towards the wing tip and thus postpone vortex bursting.
The extension of vortex stretching is more explicit when a decomposition into sub-terms, i.e.
$[S^{(+)}]$
and
$[S^{(-)}]$
(integrations inside the
$\varSigma _1$
boundary), is displayed (figure 13
c). For the rectangular wing, as the spanwise flow develops (
$\lambda$
= 2), the inboard region is dominated by the LEV stretching (
$[S^{(-)}]$
), and this benefit is sustained until the steady state (
$\lambda$
= 6). Beyond this region, an adverse vortex compression (
$[S^{(+)}]$
) is generated within
$|z/c|$
= 2–3, corresponding to a transform into vortex bursting. Unlike spanwise convection, there is almost no overlapping region between remarkable (
$[S^{(+)}]$
) and (
$[S^{(-)}]$
) regions. For the linear swept wing with
$\beta$
= 40
$^{\circ }$
, although the vortex stretching (
$[S^{(-)}]$
) is also remarkable at the same time instance (
$\lambda$
= 2), it covers a significantly wider spanwise range (
$|z/c|$
= 1.2–2.7). Thus, the outboard vortex compression is suppressed further towards the wing tip. The nonlinear swept wing with
$\Delta c$
= 1.26 also experiences similar variations, but the extension of vortex stretching is not comparable to that on the linear swept wing. In addition, the vortex compression on the swept wings is remarkably enhanced in strength (figure 13
c), indicating a more abrupt conversion from a compact LEV into a bursting LEV.
Our results thus support that the spanwise incoming flow gradient led by the swept leading edges can enhance both the spanwise vorticity convection and vortex stretching, thus retaining the LEV stability. More importantly, these results prove that a nonlinear sweep angle is not necessary to manipulate the strength of vortex stretching on the revolving wing. In addition to Wong et al. (Reference Wong, Kriegseis and Rival2013), our research further shows a spanwise gradient of incident velocity, without a concomitant change of effective incidence, is sufficient to promote LEV stability.
3.4. Lift generation analysis
For natural flapping wings and simplified revolving wings, their lift generation is dictated by the LEV evolution. As a result, the lift of three representative cases is compared in this section. As shown in figure 14(a), apart from the initial peak of the added-mass force, the transient lift generated by both swept wings is slightly lower than the rectangular wing until the end of motion. The lift loss is more prominent in the LEV formation (
$\lambda$
= 0–4), and the linear swept wing with
$\beta$
= 40
$^{\circ }$
experiences a higher reduction due to a lower LEV strength (figure 7
a). These results imply that improving outboard LEV stability and avoiding LEV bursting by enhancing vorticity transport may not be beneficial to lift generation. This may be related to the excessive vorticity removal from the inboard region, which leads to unexpected loss of local lift.
Aerodynamic force generation: (a) transient lift, (b) spatio-temporal evolution of sectional lift and (c) pressure on the dorsal surface. The correspondence between panels (b) and (c) is shown by lines.
The evolution of sectional lift on three typical cases is summarised in figure 14(b). In general, all typical cases show a similar spatio-temporal evolution, in that strong lift is generated near the wing tip during
$\lambda$
= 0–2 and is then shifted towards the mid-span. However, remarkable lift is generated on the swept wings with
$|z/c|$
= 2.5–3.5 and
$\lambda$
= 1–2, which is not observed on the rectangular wing. This is because the compact LEV is stable along the span during
$\lambda$
= 0–1 (figure 5). Thus, it achieves the highest strength near the wing tip and induces a remarkable suction force (
$\lambda$
= 1, figure 14
c). As the wing revolves, the strong low-pressure region on swept wings shifts to the spanwise position where the swept leading edges start. This is attributed to the transient circulation enhancement led by the spanwise projection of incoming velocity and thus enhanced inboard vorticity convection (figures 5 and 7
b).
As the LEV grows, the low-pressure region is mostly confined within the mid-span of all typical wings (
$\lambda$
= 4–6, figure 14
c). Note that, due to the formation of an attached TEV, another low-pressure region is induced at the trailing edge of the linear swept wing, but its contribution to the overall lift is limited, which is not sufficient to overcome the lift loss due to a weaker LEV. This proves that the degradation of the low-pressure region in the stable LEV region dominates the lift loss of swept wings.
4. Conclusion
The LEV on revolving wings can encounter bursting and unsteady shedding when the local radius exceeds certain limits. Inspired by previous hypotheses, the rectangular revolving wing is modified by swept leading edges in the outboard region to enhance the spanwise flow and thus improve LEV attachment. Both linear and nonlinear profiles are imposed, and the flow data are obtained via an in-house numerical solver. The Reynolds number, aspect ratio and angle of attack are fixed at 1500, 3 and 45
$^{\circ }$
, respectively. Results show that a swept profile can enhance the spanwise flow beyond the mid-span, but this improvement is more prominent during the LEV formation period. Moreover, the enhanced spanwise flow can intensify both vorticity convection and vortex stretching in the spanwise direction. The effective region of both vorticity transport terms can be extended further towards the wing tip, accompanied by an increase in intensity. It is thus proved that swept leading edges can reduce the inboard LEV strength and improve LEV stability in the outboard region of revolving wings via enhanced vorticity transport. It is further shown that, given an expected deflection at the wing tip, a linear profile can slightly outperform a nonlinear one in terms of further enhanced spanwise flow and vorticity transport. In addition to the conclusions of Wong et al. (Reference Wong, Kriegseis and Rival2013), our results proved that a spanwise gradient of incident velocity itself can guarantee a regulation of LEV intensity via vorticity transport, and a spanwise gradient of incident angle is not necessary.
The changes in flow features can reshape the LEV structure on swept wings, and one of the most remarkable changes is the connection of LEV with the tip vortex. In addition, the swept wings can generate an attached TEV in the swept part of the wings, which otherwise is detached on the rectangular wing. The TEV attachment is maintained by the vorticity convection and can induce an additional suction force on the dorsal surface. However, since the swept wings reduce the LEV strength, the intensity of the suction force near the leading edge is attenuated. Despite an extended stable LEV along the span and the additional suction force near the trailing edge, the lift of swept wings is inferior to the rectangular wing. This lift loss is more obvious during the LEV formation period due to the remarkably enhanced spanwise flow and vorticity transport. This presents an important finding that improving LEV attachment by limiting its circulation on swept wings may not result in a lift enhancement. Thus, potential flow control methods for lift enhancement on revolving wings should not only shift the LEV bursting towards the wing tip but also ensure a fully developed LEV.
Funding
The authors acknowledge the support from the National Natural Science Foundation of China (Nos. 12472231 and 12072013) and the China Postdoctoral Science Foundation (Nos. BX20220368).
Declaration of interests
The authors report no conflict of interest.
Author contributions
L. Chen: Conceptualisation (equal); Methodology; Investigation; Funding acquisition; Writing – original draft. J. Wu: Conceptualisation (equal); Supervision; Resources; Project administration. Y. Zhang: Conceptualisation (equal); Project administration; Writing – review and editing.
Data availability statement
The data are available from the corresponding author upon reasonable request.
