On the wake and flapping dynamics of different aspect ratio flags

TL;DR

This study addresses how aspect ratio governs wake and flapping dynamics in post-critical flutter of rectangular flags. Using time- and space-resolved deformation measurements, PIV wake analyses, and synchronized drag data across 48 flags, it reveals traveling deformation waves with a nearly constant dimensionless wavelength $\lambda/L \approx 1.87$ and shows that edge effects reduce the spanwise dynamic pressure, lowering wave speeds as $H/L$ decreases. The temporal scales (wave speed, frequency) and wake structures scale with mass ratio $M^*$ and aspect ratio, and vorticity shedding and circulation collapse when scaled by $L^*$ or $\sqrt{LH}$, highlighting three-dimensional edge effects. A semi-empirical drag model based on tip velocity and $M^*$, $St_A$, accurately predicts the mean drag ${\bar{C}}_x$ across the studied range, linking deformation, flow, and forces in a unified framework with potential applications in energy harvesting and biomedical devices.

Abstract

The flapping of flags is a classical problem involving fast and large amplitude deformations of a thin flexible plate and unsteady flow phenomena. We perform systematic time and space-resolved measurements of the deformation and drag acting on flapping flags for various aspect ratios and mass ratios. Bending waves travel from the root to the tip at a speed close to the incoming flow and the typical wavelength of the waves scales with the length of the flag. With smaller aspect ratio, the local dynamic pressure exerted by the fluid on the flag is reduced, lowering the wave propagation speed, and reducing the tip frequency. The effect of aspect ratio on the vortex formation in the near wake is analysed using flow field measurements. We identified two characteristic length scales, the ratio of the flag area over its perimeter $L^*$ and the square root of area $\sqrt{HL}$ that scale the circulation shed during a cycle. Changes in aspect ratio and mass ratio generate a wide scattering of the mean drag coefficient, ranging from 0 to 0.55. We discuss a kinematic-based model for the mean drag coefficient. This model uses the mass ratio and the typical tip speed, which depends linearly on the wave speeds, to predict the mean drag coefficient without any fitting parameter.

Figures (10)

• Figure 1: (a) Photograph of the experimental setup. (b) Sketch of the flag with its centreline highlighted with $s=0$ at the root and $s=L$ at the tip. (c) Exemplary visualisation of the time-resolved measurements of the deformation of the centreline.
• Figure 2: (a) Dimensionless tip amplitude $A/L$ and (b) dimensionless flapping frequency $fL/{\textrm{U}}_{\hbox{$\infty$}}$ as a function of reduced velocity $U^*$ for a flag of length $L=\qty{16}{cm}$ and height $H=\qty{14.5}{cm}$ corresponding to $H/L = 0.91$. The vertical dashed line in (b) represents the reduced offset velocity. (c) Reduced offset velocities ${U}_{\hbox{offset}}^*$ for the 48 tested flags compared with instability thresholds from the analytical models of argentina_fluid-flow-induced_2005shelley_heavy_2005 for infinite aspect ratio flags and from eloy_flutter_2007 for flags with three aspect ratios.
• Figure 3: (a) Superposition of the time-resolved centreline deformations for a flag of $M^* = 2.27$ and $H/L = 0.91$ with individual deformations highlighted by the darker lines. The two arrows indicate the position of the two necks for this double flutter mode. (b) Superposition of the time-resolved transverse deformation along the curvilinear abscissa $s$. (c, e) Envelope of the flapping amplitude in dimensionless coordinates $A(s)/L$ as a function of $s$. (d, f) Spatial gradient of the envelope $dA/ds$. (g) Dimensionless tip amplitude $A/L$ for each flag plotted against mass ratio and coloured by aspect ratio. For each flag, the amplitude is the averaged of measurement at different post-critical velocities and the error bar indicates the standard deviation. (h, i) Phase-averaged evolution of the time coefficients $({a}_{\hbox{1}},{a}_{\hbox{2}},{a}_{\hbox{3}})$ of the three first beam modes. Flags of same mass ratio $M^* = 2.27$ and different aspect ratio are compared in (c, d, and h). Flags of similar aspect ratio $H/L \approx 1$ and different mass ratio are compared in (e, f, and i).
• Figure 4: (a, b) Space-time diagrams of the lateral displacement $y/L$ for 10 convective times for two flags (a) $H/L=0.28$ and (b) $H/L=0.91$, both at $M^*=2.27$. (c) Correlation delay along the curvilinear coordinate for the data shown in (a) and (b). The slope in the shaded area corresponding to $s>{r}_{\hbox{2}}$ is extracted to get the corresponding wave speed $c$ and wavelength $\lambda = c/f$. (d) Wave length $\lambda/L$ as a function of aspect ratio for all tested flags. The dashed horizontal lines indicate the dimensionless wavelengths of the three first cantilevered beam modes.
• Figure 5: (a) Dimensionless wave speed $c/{\textrm{U}}_{\hbox{$\infty$}}$ obtained from the correlation delay plots against aspect ratio $H/L$, coloured by mass ratio $M^*$. (b) Estimate of the overall dynamic pressure difference $\|<\bar{p}_n>_{\bar{z}}\|_{s,t}$ against aspect ratio compared with the correction factor $A(H/L)$ from argentina_fluid-flow-induced_2005 and the spanwise averaged pressure correction ${<\tilde{p}>}_{\hbox{z}}$ from eloy_flutter_2007. (c) Experimental values of the amplitude-based Strouhal number ${St}_{\hbox{A}} = 2fA/{\textrm{U}}_{\hbox{$\infty$}}$ (circles) as a function of aspect ratio. For comparison, we included results from three-dimensional numerical simulations at $Re=500$ (rectangles) and the infinite aspect-ratio limit (dashed line) from huang_three-dimensional_2010. (d) Experimental values of the length-based Strouhal number $fL/{\textrm{U}}_{\hbox{$\infty$}}$ (circles) as a function of aspect ratio. For comparison, we included experimental results from virot_fluttering_2013 (triangles). (e) Dimensionless wave speed $c/{\textrm{U}}_{\hbox{$\infty$}}$ against flapping frequency $fL/{\textrm{U}}_{\hbox{$\infty$}}$. The dashed line indicates the linear relationship between $c/{\textrm{U}}_{\hbox{$\infty$}}$ and $fL/{\textrm{U}}_{\hbox{$\infty$}}$ for the mean measured dimensionless wavelength $\lambda/L=1.87$.