Vortex coupling in trailing vortex-wing interactions

The interaction of trailing vortices of an upstream wing with rigid and flexible downstream wings has been investigated experimentally in a wind tunnel, using particle image velocimetry, hot-wire, force, and deformation measurements. Counter-rotating upstream vortices exhibit increased meandering when they are close to the tip of the downstream wing. The upstream vortex forms a pair with the vortex shed from the downstream wing and then exhibits large displacements around the wing tip. This coupled motion of the pair has been found to cause large lift fluctuations on the downstream wing. The meandering of the vortex pair occurs at the natural meandering frequency of the isolated vortex, with a low Strouhal number, and is not affected by the frequency of the large-amplitude wing oscillations if the downstream wing is flexible. The displacement of the leading vortex is larger than that of the trailing vortex; however, it causes highly correlated variations of the core radius, core vorticity, and circulation of the trailing vortex with the coupled meandering motion. In contrast, co-rotating vortices do not exhibit any increased meandering.

Figure 1

(a) Experimental setup in the open section wind tunnel; (b) downstream view of both wings at 30% chord and definitions of wing separations.

Figure 2

Time-averaged vorticity patterns in the crossflow plane at $x/c=0.05$, for ${\alpha }_{\mathrm{LW}}={10}^{\circ },{\alpha }_{\mathrm{TW}}=5^{\circ }$. (a) $\mathrm{\Delta }y/c=-0.3$ and $\mathrm{\Delta }z/c=0.3$; (b) $\mathrm{\Delta }y/c=-0.5$ and $\mathrm{\Delta }z/c=0$; (c) $\mathrm{\Delta }y/c=0$ and $\mathrm{\Delta }z/c=-0.3$; (d) $\mathrm{\Delta }y/c=0$ and $\mathrm{\Delta }z/c=0.2$; (e) $\mathrm{\Delta }y/c=0$ and $\mathrm{\Delta }z/c=0.1$; (f) $\mathrm{\Delta }y/c=0$ and $\mathrm{\Delta }z/c=0$.

Figure 3

Three instantaneous velocity fields in the crossflow plane at $x/c=0.05$, for ${\alpha }_{\mathrm{LW}}={10}^{\circ },{\alpha }_{\mathrm{TW}}=5^{\circ },\mathrm{\Delta }y/c=0$, and $\mathrm{\Delta }z/c=0.1$.

Figure 4

Time-averaged vorticity patterns in the crossflow plane at $x/c=0.05$, for ${\alpha }_{\mathrm{LW}}=-{10}^{\circ },{\alpha }_{\mathrm{TW}}=5^{\circ }$. (a) $\mathrm{\Delta }y/c=0$ and $\mathrm{\Delta }z/c=-0.1$; (b) $\mathrm{\Delta }y/c=0$ and $\mathrm{\Delta }z/c=-0.2$; (c) $\mathrm{\Delta }y/c=0$ and $\mathrm{\Delta }z/c=-0.3$; (d) $\mathrm{\Delta }y/c=-0.4$ and $\mathrm{\Delta }z/c=0$.

Figure 5

Time-averaged vorticity patterns in the crossflow plane at $x/c=0.05$, for ${\alpha }_{\mathrm{TW}}={10}^{\circ }$. (a) ${\alpha }_{\mathrm{LW}}={10}^{\circ },\mathrm{\Delta }y/c=-0.1$, and $\mathrm{\Delta }z/c=0.2$; (b) ${\alpha }_{\mathrm{LW}}={10}^{\circ },\mathrm{\Delta }y/c=0$, and $\mathrm{\Delta }z/c=0.1$; (c) ${\alpha }_{\mathrm{LW}}={10}^{\circ },\mathrm{\Delta }y/c=0$, and $\mathrm{\Delta }z/c=-0.1$; (d) ${\alpha }_{\mathrm{LW}}=-{10}^{\circ },\mathrm{\Delta }y/c=0$, and $\mathrm{\Delta }z/c=-0.25$; (e) ${\alpha }_{\mathrm{LW}}=-{10}^{\circ },\mathrm{\Delta }y/c=0$, and $\mathrm{\Delta }z/c=-0.1$; (f) ${\alpha }_{\mathrm{LW}}=-{10}^{\circ },\mathrm{\Delta }y/c=0$, and $\mathrm{\Delta }z/c=0.1$.

Figure 6

Time-averaged vorticity patterns in the crossflow plane at $x/c=0.05$, for ${\alpha }_{\mathrm{LW}}=5^{\circ },{\alpha }_{\mathrm{TW}}=5^{\circ }$. (a) $\mathrm{\Delta }y/c=0$ and $\mathrm{\Delta }z/c=0.15$; (b) $\mathrm{\Delta }y/c=0$ and $\mathrm{\Delta }z/c=0.05$; (c) $\mathrm{\Delta }y/c=0$ and $\mathrm{\Delta }z/c=-0.05$.

Figure 7

Percent change in time-averaged lift (top) and in root-mean square lift (bottom) as a function of wing separations in the spanwise and normal directions, ${\alpha }_{\mathrm{LW}}={10}^{\circ },{\alpha }_{\mathrm{TW}}=5^{\circ }$. The baseline case is the trailing wing in the freestream. The horizontal solid line represents the trailing-edge of the wing.

Figure 8

Three-dimensional view of time-averaged vorticity patterns in the crossflow planes from $x/c=-1.05$ to 0.05; (a) ${\alpha }_{\mathrm{LW}}={10}^{\circ },{\alpha }_{\mathrm{TW}}=5^{\circ },\mathrm{\Delta }y/c=0$, and $\mathrm{\Delta }z/c=0.1$; (b) ${\alpha }_{\mathrm{LW}}={10}^{\circ },{\alpha }_{\mathrm{TW}}={10}^{\circ },\mathrm{\Delta }y/c=0$, and $\mathrm{\Delta }z/c=0.1$; (c) ${\alpha }_{\mathrm{LW}}={10}^{\circ },{\alpha }_{\mathrm{TW}}={10}^{\circ },\mathrm{\Delta }y/c=-0.1$, and $\mathrm{\Delta }z/c=0.2$.

Figure 9

Time-averaged vorticity patterns and flow structures of the first two POD modes in the crossflow plane at $x/c=0.05$, leading vortex alone, ${\alpha }_{\mathrm{LW}}={10}^{\circ }$.

Figure 10

Frequency spectra of velocity near the core of the isolated vortex at $x/c=0.05,{\alpha }_{\mathrm{LW}}={10}^{\circ }$, hot-wire probe location $y/c=0.2,z/c=0$.

Figure 11

Time-averaged vorticity patterns and flow structures of the first two POD modes in the crossflow planes (a) $x/c=-1.05$ (left), (b) $x/c=0.05$ (right). ${\alpha }_{\mathrm{LW}}={10}^{\circ },{\alpha }_{\mathrm{TW}}=5^{\circ },\mathrm{\Delta }y/c=0$, and $\mathrm{\Delta }z/c=0.1$.

Figure 12

Frequency spectra of velocity near the vortex core in the planes at $x/c=-1.05$ (upstream) and 0.05 (downstream), for ${\alpha }_{\mathrm{LW}}={10}^{\circ },{\alpha }_{\mathrm{TW}}=5^{\circ },\mathrm{\Delta }y/c=0$, and $\mathrm{\Delta }z/c=0.1$. Hot-wire probe location $y/c=0.2,z/c=0$.

Figure 13

Probability of instantaneous location of the leading vortex in the crossflow plane at $x/c=-1.05$ (top) and $x/c=0.05$ (bottom), ${\alpha }_{\mathrm{LW}}={10}^{\circ },{\alpha }_{\mathrm{TW}}=5^{\circ },\mathrm{\Delta }y/c=0$, and $\mathrm{\Delta }z/c=0.1$.

Figure 14

Time histories of spanwise and normal locations of the leading (red) and trailing (black) vortices in the crossflow plane at $x/c=0.05$. ${\alpha }_{\mathrm{LW}}={10}^{\circ },{\alpha }_{\mathrm{TW}}=5^{\circ },\mathrm{\Delta }y/c=0$, and $\mathrm{\Delta }z/c=0.1$.

Figure 15

Time-averaged vorticity patterns in the crossflow plane at $x/c=0.05$, for ${\alpha }_{\mathrm{LW}}={10}^{\circ },{\alpha }_{\mathrm{TW}}=5^{\circ }$, flexible trailing wing: (a) $\mathrm{\Delta }y/c=0.4$ and $\mathrm{\Delta }z/c=0.6$; (b) $\mathrm{\Delta }y/c=0$ and $\mathrm{\Delta }z/c=0.6$; (c) $\mathrm{\Delta }y/c=-0.4$ and $\mathrm{\Delta }z/c=0.6$. Dashed lines indicate mean location of the trailing wing; “+” signs in (b) indicate the standard deviation of deformation.

Figure 16

Percent change in time-averaged lift (top) and in root-mean square lift (bottom) of the flexible wing as a function of wing separations in the spanwise and normal directions, ${\alpha }_{\mathrm{LW}}={10}^{\circ },{\alpha }_{\mathrm{TW}}=5^{\circ }$. The baseline case is the trailing wing in the freestream.

Figure 17

Three-dimensional view of time-averaged vorticity patterns in the crossflow planes from $x/c=-1.05$ to 0.05; ${\alpha }_{\mathrm{LW}}={10}^{\circ },{\alpha }_{\mathrm{TW}}=5^{\circ },\mathrm{\Delta }y/c=0$, and $\mathrm{\Delta }z/c=0.6$, flexible trailing wing.

Figure 18

Time-averaged vorticity patterns (top), probability of instantaneous location of leading vortex (middle), and frequency spectra of the velocity (hot-wire probe location $y/c=0.2,z/c=0.5)$ and wing-tip deflection at midchord in the upstream plane $x/c=-1.05$ (left) and downstream plane $x/c=0.05$ (right). ${\alpha }_{\mathrm{LW}}={10}^{\circ },{\alpha }_{\mathrm{TW}}=5^{\circ },\mathrm{\Delta }y/c=0$, and $\mathrm{\Delta }z/c=0.6$.