Versatile reduced-order model of leading-edge vortices on rotary wings

The leading-edge vortex (LEV) is a universal and robust lift enhancement mechanism in biological flapping and autorotating flight. It is characterized by comparatively low Reynolds number and large angle of attack leading to separated three-dimensional flow, which has long precluded analytical approach to this problem. Here we propose a reduced-order analytical model, which is capable of delivering a fast closed-form estimate of the LEV strength and position for a revolving wing at arbitrary angle of attack. It is postulated that equilibrium state of the vortex is primarily maintained by the competing effects of vorticity production at the leading edge and its three-dimensional transport in the presence of spanwise flow and downwash. The results of the model are found consistent with the LEV strength and centroid coordinates measured previously in experiments, as well as determined from numerical solution of the Navier-Stokes equations, in a wide range of the Reynolds number. The agreement is good not only for rectangular wings, but also for bio-inspired shapes of fruit fly, bumblebee, and hawkmoth. Hence, the model offers a simple, versatile, and reliable tool for practical estimation of the LEV parameters.

Figure 1

(a) Flow visualization of a rectangular flat-plate wing that rotates about its root with constant angular velocity $\mathrm{\Omega }$. $R$ is the distance from the axis of rotation to the tip edge; $c$ is the chord length. The angle of attack $\alpha$ is defined as $\pi /2$ minus the angle between the rotation axis and the root edge. Vortices in the surrounding fluid are visualized using an isosurface of the ${\lambda }_2$ criterion, colored according of the sign of spanwise vorticity component. The red cone structure above the upper surface of the wing is the leading-edge vortex (LEV). (b) Line-vortex approximation of the LEV. A Lagrangian vortex element that originates from the leading edge of the root section is transported towards the wing tip with the spanwise velocity $V_r$. Simultaneously, it is swept away from the leading edge moving through the surrounding fluid with local velocity $\mathrm{\Omega }r$. Circulation ${\mathrm{\Gamma }}_1$ of the vortex increases with $r$ by the amount of vorticity produced at the leading edge. Spanwise variation of the LEV circulation, including its termination at the wing tip, is accompanied by chordwise vorticity, which induces downwash $w_i$. Subscripts “I,” “II,” and “III” denote the velocities sampled on three different cylindrical planes of radius $r_{\mathrm{I}},r_{\mathrm{II}}$, and $r_{\mathrm{III}}$, respectively. (c) Two-dimensional approximation is used to describe the flow on a cylindrical plane. Spanwise evolution of the vortex is substituted with time evolution on a cylinder plane of radius that varies with time $\tau$ [Eq. (1)]. Inflow velocity $V$ is calculated as vector sum of $\mathrm{\Omega }r$ and $w_i$. Circulation ${\mathrm{\Gamma }}_1$ and position $z_1=x_1+i{y}_1$ of the vortex on a complex plane are calculated using conformal mapping between a slit plane and a half-plane. Polar coordinates ${\rho }_1$ and ${\theta }_1$ are used for describing the position of the vortex and its mirror image on the preimage plane $\zeta$, using the coordinate transform $\zeta =\rho exp\left[i\right(\pi /2-\theta \left)\right]$. (d) Spanwise profiles of the normalized LEV circulation (14) of a wing with aspect ratio $R/c=4$ at Reynolds number $\text{Re}=\mathrm{\Omega }Rc/\nu =500$, evaluated at five different angles of attack $\alpha$. (e) Normalized perpendicular, $x_1$, and parallel, $y_1$, coordinates of the vortex centroid with respect to the leading edge; see (13).

Figure 2

Normalized values of (a) circulation $\mathrm{\Gamma }$ and (b) perpendicular, $x$, and parallel, $y$, coordinates of the vortex with respect to the leading edge, as functions of the angle of attack $\alpha$, at a constant Reynolds number $\text{Re}=\mathrm{\Omega }Rc/\nu =500$. (c) LEV circulation as a function of Re, at $\alpha ={45}^{\circ }$. Continuous lines correspond to the theoretical estimates (14) for ${\mathrm{\Gamma }}_1$ and (13) for $z_1$, evaluated at spanwise stations $r=0.5c,c$, and $2c$, which correspond to $12.5\%,25\%,$ and $50\%$ of the wing length $\left(R=4c\right)$, respectively. Discontinuous lines show the normalized circulation obtained from the CFD simulations. Dotted, dash-dot, and dashed lines correspond to, respectively, $\varphi ={84}^{\circ },{180}^{\circ }$, and ${270}^{\circ }$ angular distance travelled by the wing. Inset shows the half-cylinder surface $r=\text{const},0\le y^{\prime }\le L_y/2,0\le {\varphi }^{\prime }<2\pi$ used as domain of integration of the radial vorticity ${\omega }_r$, which yields $\mathrm{\Gamma }={\int }_0^{2\pi }{\int }_0^{L_y/2}{\omega }_{r}rd{y}^{\prime }d{\varphi }^{\prime },x+iy=\frac{1}{\mathrm{\Gamma }}{\int }_0^{2\pi }{\int }_0^{L_y/2}(r{\varphi }^{\prime }+i{y}^{\prime }){\omega }_{r}rd{y}^{\prime }d{\varphi }^{\prime }$. $L_y$ is the CFD domain height.

Figure 3

Comparison between the theory (black lines), PIV measurements [7] (red markers), and CFD results (blue markers) at the rotation angle $\varphi ={108}^{\circ }$. (a) Dimensionless LEV circulation and (b) LEV centroid position relative to the leading edge, viewed from above in the direction of the axis of rotation. Normalization is as in Ref. [7], where $b$ is the wing length from the root to the tip, $r_{\mathrm{root}}$ is the distance from the axis of rotation to the wing root, and $U_{\mathrm{tip}}=\mathrm{\Omega }(r_{\mathrm{root}}+b)$. The domain of integration for calculating $\mathrm{\Gamma }$ is shown in the inset. (c) Circulation normalized with $\mathrm{\Omega }{c}^2$ and (d) position relative to the leading edge, plotted as functions of the dimensionless distance from the axis of rotation $r/c$. The dash-dot line in panel (c) corresponds to $2{(r/c)}^{4/3}$. Panels (b) and (d) show the view from the top in the direction parallel to the rotation axis such that $(ycos\alpha -xsin\alpha )/c=0$ corresponds to the leading edge and $-0.707$ to the trailing edge, as required for comparison with the results of the PIV measurement shown in Ref. [7].

Figure 4

Estimates of the spanwise vorticity transport coefficient $K_{90}$ as a function of the root-based Reynolds number ${\text{Re}}_1=\mathrm{\Omega }{c}^2/\nu$. The values shown with green circles best fit the theoretical estimate to the results of a CFD model with aspect ratio 6; see Ref. [15]. Blue squares depict similar results for a wing of aspect ratio 4, in a wider range of ${\text{Re}}_1$. The dashed line corresponds to the empirical formula (16).

Figure 5

(a) Chordwise component of the LEV centroid position versus the spanwise distance from the axis of rotation, for three different values of the root-based Reynolds number ${\text{Re}}_1$. Intersection with the dash-dot line plotted at $y/c=0.5$ determines the critical spanwise distance $r_{\mathrm{crit}}$. All distances are normalized with $c$. The view is in the direction perpendicular to the wing surface such that $y/c=0$ corresponds to the leading edge and $-1$ to the trailing edge. (b) $r_{\mathrm{crit}}$ plotted as a function of the spanwise vorticity transport parameter $K_{90}$, the latter being varied in a hypothetical range from 0.1 to 1. The wing has the aspect ratio $R/c=10$, and it operates at $\alpha ={45}^{\circ }$.

Figure 6

The effect of the downwash parameter $\kappa$ on the LEV (a) circulation and (b) centroid position, shown for $\text{Re}=500,R/c=4,r/c=2$, and $\alpha$ varied between 0 and ${90}^{\circ }$.

Figure 7

(a) Fruit fly, (b) bumblebee, and (c) hawkmoth LEVs, at their respective Re equal to 100, 2800, and 5400, at $\alpha ={40}^{\circ }$. Dimensionless circulation $\mathrm{\Gamma }/\mathrm{\Omega }{c}^2$ is plotted as a function of dimensionless distance from the rotation axis, $r/c$, where $c$ is the mean chord length for each species. Continuous, dotted, and dashed lines show the CFD data, linear fit to it, and the theoretical estimates, respectively. A reference line $2{(r/c)}^{4/3}$, plotted with dash-dot, is displayed for the ease of comparison between the different cases. (d) The three wing contours, normalized by $c$ (black horizontal scale bar), superposed, and having the same pivot point (black marker).