Mode competition in a plunging foil with an active flap: A multiscale modal analysis approach

Flow-induced fluttering has a significant role in aircraft stability, renewable energy extraction, animal locomotion, and many other applications. While being a ubiquitous phenomenon, the control of the flutter response has been primarily limited to simplified systems and, often, with the help of linear inviscid flow theories. In this paper, we numerically investigate how the plunging response of a foil can be regulated using an active flap to improve structural safety or enhance the energy extraction efficiency of the foil with a tightly coupled fluid-structure interaction algorithm. A broad range of foil and flap settings was tested, and their flow dynamics have been investigated. A multiscale modal analysis technique suitable for fluid-structure interaction systems is employed to systematically isolate the active flap-induced and flow-induced modes. It is observed that the competition between these two modes dictates the plunging response of the foil. The active flap can modulate the leading edge vortex shedding with larger flapping amplitude and regulate the foil heaving motion. The ratio of the competing modal energy is proposed to evaluate the control efficacy of the morphing surface, and the onset of the lock-in is associated with the ratio approaching unity. It is shown that the morphing flap is a good candidate for active flow control.

Figure 1

(a) The cross section of the foil and the model parameters. (b) Sample computational grid around the foil-flap model and the grid mapping to a unit circle.

Figure 2

(a) Dominant frequency of the heaving motion, (b) mean heaving displacement, and (c) rms of heaving amplitude of the uncontrolled cases.

Figure 3

(a) Dominant nondimensional heaving frequency, (b) mean heaving displacement, and (c) unsteady heaving amplitude (rms) for $\mathrm{Ao}\mathrm{A}=0^{\circ }$ and different flap positions ${\theta }_0$ and flap frequency ${\text{St}}_f$. Flap amplitude is fixed at $A/{\theta }_0=1$.

Figure 4

(a) Dominant nondimensional heaving frequency, (b) average heaving displacement, and (c) heaving amplitude (rms) for $\mathrm{Ao}\mathrm{A}={10}^{\circ }$ with flap amplitude $A=1.5^{\circ }$ at different average angle ${\theta }_0$.

Figure 5

The (a) heaving motion time histories, (b) power spectra, and (c) the phase portraits of ${\text{St}}_f=0.0,0.1,0.9,1.2$. It is assumed that $\mathrm{Ao}\mathrm{A}={10}^{\circ }, {\theta }_0={24}^{\circ }$, and $A=1.5^{\circ }$.

Figure 6

Instantaneous vorticity fields over four active flap periods of ${\text{St}}_f=0.1$ and ${\text{St}}_f=0.9$. $T$ is the flap motion period.

Figure 7

(a) Dominant nondimensional heaving frequency, (b) mean heaving displacement, and (c) unsteady heaving amplitude (rms) for $\mathrm{Ao}\mathrm{A}={10}^{\circ }$ with the large-amplitude flap motion of $A/{\theta }_0=0$.

Figure 8

The heaving displacement time history, power spectrum, and phase portrait plot of ${\text{St}}_f=0.0,0.1,1.0,1.1$, respectively, from the top to bottom row. The results are for $\mathrm{Ao}\mathrm{A}={10}^{\circ }$ and $A={\theta }_0={24}^{\circ }$.

Figure 9

Instantaneous vorticity field over four active flap periods of (a) ${\text{St}}_f=1.0$ and (b) ${\text{St}}_f=1.1$ with $\mathrm{Ao}\mathrm{A}={10}^{\circ }$ and flap motion $A={\theta }_0={24}^{\circ }$.

Figure 10

(a) The discrete Fourier transform (DFT) of the normalized temporal correlation matrix; (b) partitioned correlation matrix. (c) The spatial structure and spectrum of the leading modes of the case $\mathrm{Ao}\mathrm{A}={10}^{\circ }$ with flap frequency ${\text{St}}_f=0.1$, amplitude $A=1.5^{\circ }$, and average angle at ${\theta }_0={24}^{\circ }$. The modes are normalized based on their respective largest vorticity, which is the same for the following figures.

Figure 11

The spatial structures and spectra of the leading flap-induced and flow-induced modes of the case $\mathrm{Ao}\mathrm{A}={10}^{\circ }$ with flap amplitude $A=1.5^{\circ }$, average angle ${\theta }_0={24}^{\circ }$, and different flap frequency ${\text{St}}_f$.

Figure 12

The spatial structures and spectra of the flap-induced and flow-induced modes of the case $\mathrm{Ao}\mathrm{A}={10}^{\circ }$ with flap amplitude $A={\theta }_0={24}^{\circ }$ and different flap frequency ${\text{St}}_f$. The modes are normalized based on their respective largest vorticity.

Figure 13

Ratio of the leading flap-induced and flow-induced modal energy of $\mathrm{Ao}\mathrm{A}={10}^{\circ }$ case with (a) small flap amplitude $A=1.5^{\circ }$ and average angle ${\theta }_0={24}^{\circ }$ and (b) large amplitude $A={\theta }_0={24}^{\circ }$. See text for explanation of the Roman numbers.

Figure 14

(a) Instantaneous snapshot of the vorticity field with the resolution of $140\times 128, 280\times 256$, and $560\times 512$ in the radial and azimuthal directions, respectively. (b) Comparison of the heaving time histories with three different grid resolutions.

Figure 15

The average heaving displacement and heaving amplitude of (a) varied stiffness with fixed unity damping and (b) varied damping with fixed unity stiffness.