Spectral energy cascade of body rotations and oscillations under turbulence

The rotation of rigid bodies and oscillation of flexible structures under turbulence are described and characterized in the spectral domain. Laboratory experiments combined with theory are used to unravel and explain the characteristic power-law decay of the spectrum of rotation and oscillation of simple structures. It is shown that the energy-containing eddies and inertial subrange of the incoming velocity spectrum strongly modulate the unsteady motions of structures. The spectra of the rotation and oscillation of the structures exhibit distinctive power-law decays $f^{+2}$ and $f^{-5/3-2}$ in the comparatively low- and high-frequency regions. The range of these regions depends on the phenomenon triggering the motions, i.e., thrust or vortex-induced motions.

Figure 1

Schematic of the experimental setup for the (a) rotations and (b) oscillations of plates. The parameters are $x\times y=50\times 150$ and $85\times 85{\mathrm{mm}}^2$ and $H=80$, 100, and 150 mm.

Figure 2

Spectra of the rotations ${\mathrm{\Phi }}_{\omega }$ of various structures: (a) rectangular plate of $b/c=1/12$, (b) rectangular plate of $b/c=1/4$, (c) $\mathsf{S}$-shaped plate, and (d) model wind turbine. Note that the velocity spectrum of the incoming flow is the same in all cases and, for clarity, is only presented in (d) with blue.

Figure 3

Decoupled frequency $f_d$ as a function of mean velocity $U$ for one $\mathsf{S}$-shaped and two flat plates.

Figure 4

General schematic of the transfer function $Q\left(f\right)$ associated with the velocity oscillations of the structures.

Figure 5

Spectra of the plate oscillations ${\mathrm{\Phi }}_{\dot{\theta }}$ for two mean flow velocities: (a) rectangular and (b) square plates.

Figure 6

Comparison of the modeled and measured spectra of the plate oscillations ${\mathrm{\Phi }}_{\dot{\theta }}$: (a) spectrum of the incoming flow for $U=6$ $\mathrm{m}{\mathrm{s}}^{-1}$ case, (b) square plate with high stiffness support under $U=6$ $\mathrm{m}{\mathrm{s}}^{-1}$, (c) square plate with low stiffness support and $U=5$ $\mathrm{m}{\mathrm{s}}^{-1}$, and (d) rectangular plate with medium stiffness support under $U=7$ $\mathrm{m}{\mathrm{s}}^{-1}$.