Linear drag law for high-Reynolds-number flow past an oscillating body

An object immersed in a fast flow typically experiences fluid forces that increase with the square of speed. Here we explore how this high-Reynolds-number force-speed relationship is affected by unsteady motions of a body. Experiments on disks that are driven to oscillate while progressing through air reveal two distinct regimes: a conventional quadratic relationship for slow oscillations and an anomalous scaling for fast flapping in which the time-averaged drag increases linearly with flow speed. In the linear regime, flow visualization shows that a pair of counterrotating vortices is shed with each oscillation and a model that views a train of such dipoles as a momentum jet reproduces the linearity. We also show that appropriate scaling variables collapse the experimental data from both regimes and for different oscillatory motions into a single drag-speed relationship. These results could provide insight into the aerodynamic resistance incurred by oscillating wings in flight and they suggest that vibrations can be an effective means to actively control the drag on an object.

Figure 1

(a) In-line oscillations of a translating disk in a flow. (b) Experimental apparatus: Disks are driven to oscillate back and forth by a motor and are free to rotate through air in response to an applied force.

Figure 2

Force-speed characterization. (a) Time-averaged drag $D$ on each disk moving at steady dc speed $U$ while driven to oscillate at frequencies $f=6$, 8, and 10 Hz (blue, yellow, and red curves, respectively) and peak-to-peak amplitude $A=3.5$ cm. The inset shows data collapse for $D/w$ vs $U$. (b) Logarithm-scale plot of the same, revealing a linear drag-speed (slope 1) relationship for low $U$ and conventional quadratic scaling (slope 2) for high $U$. Arrows indicate the state for which $U$ equals the mean oscillation speed $w=2Af$.

Figure 3

Normalization reveals a universal drag-speed relationship for unsteady motions. (a) Drag coefficient $C_D=D/\frac{1}{2}\rho S{U}^2$, where $S$ is the disk area, plotted vs the speed ratio $J=U/w$. The dashed line indicates the experimentally measured value of $C_D^{*}=3.5\pm 0.1$ for steady motion and colored curves represent six cases of unsteady motions with different values of $f$. (b) Plotting $C_D$ vs $1/J$ emphasizes the unsteady regime $1/J>1$. Also shown are predictions of a quasisteady calculation (dotted curves) and an unsteady dipole-jet model (gray bands). See the text for model details.

Figure 4

Visualization of flow past stationary and oscillating disks (see movies in the Supplemental Material [16]). (a) Fluorescein dye streaklines reveal the flow (top to bottom) past a stationary disk oriented horizontally near the top of the photograph. (b) Streaklines for a flapping disk with speed ratio $J=U/w\approx 2$. (c) Close-up of an asymmetric dipole generated under the same conditions. (d) Time-exposed photograph of glass microspheres illuminated by a laser sheet reveal flow path lines for faster flapping ($J\approx 0.5$). A more symmetric dipole is generated near the disk edge and these dipoles lose their coherence further downstream.

Figure 5

A train of vortex dipoles can be modeled as a jet. (a) A plate flaps with frequency $f$, peak-to-peak amplitude $A$, and average speed $w=2Af$ in an oncoming flow of speed $U$. (b) The forward stroke generates a clockwise vortex and the backward stroke produces a counterrotating vortex. Together, a complete stroke creates a vortex pair. (c) Repeated oscillations create a train of dipoles that can be viewed as a jet. The jet is emitted straight outward for $U=0$ and curves downstream for $U>0$.

Figure 6

A train of vortex dipoles can be viewed as a continuous jet. (a) The vorticity associated with symmetric dipoles can be spread into outer regions (gray regions) that bound an irrotational core. (b) Vorticity $\omega$ and velocity $u$ profiles across the jet. Asymmetric dipoles tend to propagate and turn. A train of such dipoles can be viewed as a curving jet. (c) The inside of the jet (region 1) is of more intense vorticity than the outside (region 2) and the jet has radius of curvature $R$. (d) Vorticity $\omega$ and velocity $u$ profiles across the jet.