Unsteady aerodynamic theory and experiments of hovering membrane wings

TL;DR

The paper develops an unsteady inviscid aerodynamic theory for hovering membrane wings, incorporating variable freestream velocity and high angles of attack with small membrane camber. It derives a closed-form lift expression that combines Theodorsen's and Wagner's functions with membrane deformation modes, and then validates the model against water-tank experiments that provide membrane-kinematics data to predict lift. The results show good agreement for attached flow at aeroelastic numbers $Ae>1.7$ and moderate camber, demonstrating that the unsteady theory extends beyond traditional low-angle regimes and enables lift predictions from measured deformation in bio-inspired membranes. The findings highlight the primacy of attachment over small camber in governing predictive accuracy and suggest practical use in designing hovering membrane-wings and flexible micro air vehicles.

Abstract

We investigate the unsteady lift response of compliant membrane wings in hovering kinematics by combining analytical inviscid theory with experimental results. An unsteady aerodynamic model is derived for a compliant thin aerofoil immersed in incompressible inviscid flow of variable freestream velocity at high angles of attack. The model, representing a spanwise section of a hovering membrane wing, assumes small membrane deformation and attached flow. These assumptions are supported by experiments showing that passive membrane deformation suppresses flow separation when hovering at angles of attack up to $55^\circ$. An analytically derived expression is obtained for the unsteady lift response, incorporating the classical Wagner and Theodorsen functions and the membrane dynamic response. This theoretical expression is validated against experimental water-tank measurements that are performed on hovering membrane wings at angles of attack of $35^\circ$ and $55^\circ$. Data from membrane deformation measurements is applied to the theoretical lift expression, providing the theoretical lift response prediction for each of the available experimental scenarios. Results of the comparison show that the proposed theory accurately predicts unsteady lift contributions from membrane deformation at high angles of attack, provided the deformation remains small and the flow is attached. This agreement between inviscid theory and experimental measurements suggests that when flow separation is suppressed, the unsteady aerodynamic theory is valid well beyond the typical low angle of attack regime.

Figures (5)

• Figure 1: Experimental study and motivation: (a) Pallas's long-tongued bat (Glossophaga soricina, photo by: Gregory Basco/www.deepgreenphotography.com), (b) Drawing of the experimental setup used by Gehrke_2022 in which the membrane wing is mounted on a flapping platform. Vorticity field snapshots of (c) rigid and (d) flexible membrane wings obtained for $\hat{\alpha}=55^\circ$ at three time instances, $t/T = 0.20, 0.25$, and $0.30$, where $T$ is the full-cycle time period that includes a forward and a backward stroke. Streamlines around the (e) rigid and (f) flexible wing obtained at $t/T = 0.25$ demonstrate how passive membrane deformation reattaches the flow and reduces flow separation in hovering flapping motion at high angles of attack up to $55^\circ$.
• Figure 2: Representation of (a) the experimental membrane wing section and surrounding flowfield and (b) the theoretical model used to derive the unsteady lift coefficient acting on a flexible membrane aerofoil in variable freestream velocity. PIV results shown on the experimental model support the assumptions of attached flow and wake extension along the chord line, utilized in the theoretical model.
• Figure 3: (a) Sketch of the flexible membrane wing profile, (b) Angle of attack ($\alpha$) and local wing velocity ($U$) profiles as a function of normalised time $t/T$, (c) Membrane camberline, $y(x)$, obtained at different times ($t/T = 0$ to $0.5$), (d) Membrane Fourier coefficients, $\mathcal{F}_n$, as a function of time $t/T$, (e) Measured lift coefficients, $C_L$, as a function of time $t/T$ for a membrane and a rigid wing, (f) Finite-wing lift coefficient due to membrane deformation, $C_{L_d}$, as a function of time $t/T$; comparing the experimental measurements with the theory prediction, along with simplified model predictions assuming single-mode oscillations ($\mathcal{F}_1$), constant freestream velocity ($\overline{U}$), and a quasi-steady aerodynamic model (QS). Very good agreement is observed between the theoretical and measured lift coefficients, computed for $\hat{\alpha}=35^\circ, \textit{Ae}\xspace=2.5$, which is also captured by the single-mode approximation. However, the simplified constant $\overline{U}$ model and the quasi-steady model do not agree well with measurements.
• Figure 4: Lift coefficient, maximum membrane deformation, and flowfield snapshots obtained for varying values of aeroelastic number, $Ae$, and an angle-of-attack amplitude of (i) $\hat{\alpha}=35^\circ$; (ii) $\hat{\alpha}=55^\circ$. Vorticity field snapshots in (i.d) and (ii.f) depict the flow around the rigid wing and flexible membrane wings at $t/T=0.25$ for (i.d) $\hat{\alpha}=35^\circ, \textit{Ae}\xspace=0.85,2.3,5.2$, and (ii.f) $\hat{\alpha}=55^\circ, \textit{Ae}\xspace=0.85,1.9,5.2$. Lift coefficient due to membrane deformation results in (i.a-c) and (ii.a-e) are computed for $\textit{Ae}\xspace=3.4,2.5,1.7$, and $\textit{Ae}\xspace=5,3.4,1.9,1.7,0.95$, respectively, with experimental measurements appearing in blue and theoretical prediction based on measured deformations in red. Black lines depict the measured maximum mean membrane camber, $y_{max}$. Shaded areas represent results obtained from forward and backward strokes and solid lines depict the mean values. Note that the mean freestream in the experimental setup is zero, so there is no practical difference between forward and backward strokes. Very good agreement is observed between theory and measurement for cases of $\textit{Ae}\xspace>1.7$. Lower values of Ae result in high membrane camber that leads to flow separation near the trailing edge, as evident in the flowfield snapshots.
• Figure 5: Absolute value of the time-averaged error in prediction of the lift coefficient due to membrane deformation, $|\Delta \overline{C}_{L_d}|$, plotted against (a) aeroelastic number, Ae; (b) mean maximum camber, $\overline{y}_{max}$, for angle-of-attack amplitudes of $\hat{\alpha}=35^o, 55^o$. Small Ae values lead to considerable error in $C_{L_d}$, where $|\Delta \overline{C}_{L_d}|>0.4$, and as Ae increases the error decreases.