On the unsteady dynamics of synthetic leaves: Laboratory experiments using synchronized PIV and DIC

The unsteady 3D dynamics of various synthetic leaves and the induced turbulence are systematically studied experimentally for representative Cauchy numbers in a wind tunnel under nearly uniform incoming flows. Synchronized digital image correlation (DIC) and high-frame-rate particle image velocimetry (PIV) are employed to track the structure dynamics simultaneously and the surrounding flow field to uncover the fluid-solid interaction. A high-resolution six-axis load cell is also used to quantify the synthetic leaves' induced force and torque under various flows. The shapes of synthetic leaves inspected are representative of selected environments (e.g., calm to windy weather; tropical to temperate climate). The Cauchy number is set to resemble those observed in natural conditions. This presentation will discuss insights from synchronized PIV-DIC techniques on the synthetic leaves' distinct behavior and wake flow response. Particular emphasis is placed on characterizing flow instability and the leave shape's role in the motions and force. For this purpose, we inspected the instantaneous force and torque as well as their structure. We will also discuss the relationship between leave shapes with force and torque fluctuations linking them with the leaf motion obtained from DIC measurements. In particular, the results show that selected leaf shapes experience significantly larger and distinct force and torque fluctuations and larger pitch magnitude, as shown in Fig. 5. A shared monotonically decreasing trend of the nondimensional frequency (Strouhal number, St = fL/U) is evidenced for standard environmental conditions.

Scaling of the performance of insect-inspired passive-pitching flapping wings

Flapping flight using passive pitch regulation is a commonly used mode of thrust and lift generation in insects and has been widely emulated in flying vehicles because it allows for simple implementation of the complex kinematics associated with flapping wing systems. Although robotic flight employing passive pitching to regulate angle of attack has been previously demonstrated, there does not exist a comprehensive understanding of the effectiveness of this mode of aerodynamic force generation, nor a method to accurately predict its performance over a range of relevant scales. Here, we present such scaling laws, incorporating aerodynamic, inertial and structural elements of the flapping-wing system, validating the theoretical considerations using a mechanical model which is tested for a linear elastic hinge and near-sinusoidal stroke kinematics over a range of scales, hinge stiffnesses and flapping frequencies. We find that suitably defined dimensionless parameters, including the Reynolds number, Re , the Cauchy number, Ch , and a newly defined ‘inertial-elastic’ number, IE, can reliably predict the kinematic and aerodynamic performance of the system. Our results also reveal a consistent dependency of pitching kinematics on these dimensionless parameters, providing a connection between lift coefficient and kinematic features such as angle of attack and wing rotation.

Passive Pitching Mechanism of Three-Dimensional Flapping Wings in Hovering Flight

Flapping wings of insects can passively maintain a high angle of attack due to the torsional flexibility of wing basal region without the aid of the active pitching motion. However, the lift force generated by such passive pitching motion has not been well explored in the literature. Consequently, there is no clear understanding of how torsional wing flexibility should be designed for optimal performance. In this work, a computational study was conducted to investigate the passive pitching mechanism of flapping wings in hovering flight using a torsional spring model. The torsional wing flexibility was characterized by Cauchy number. The impacts of the inertial effect of wings were evaluated using the mass ratio. The aerodynamic forces and associated unsteady flow structures were simulated by an in-house immersed-boundary-method based computational fluid dynamic solver. A parametric study on the Cauchy number was performed with a Reynolds number of 300 at a mass ratio of 1.0, which covers a wide range of species of insect wings. According to the analysis of the aerodynamic performance, we found that the optimal lift can be achieved at a Cauchy number around 0.16, while the optimal efficiency in terms of lift-to-power ratio was reached at a Cauchy number around 0.3. All the corresponding wing pitching kinematics had a pitching magnitude around 60 degrees with slightly advanced rotation. In addition, 3D wake structures generated by the passive flapping wings were analyzed in detail. The findings of this work could provide important implications for designing more efficient flapping-wing micro air vehicles.

Flow-induced motions of flexible plates: fluttering, twisting and orbital modes

The unsteady dynamics of wall-mounted flexible plates under inclined flows was fundamentally described using theoretical arguments and experiments under various Cauchy numbers $Ca=\unicode[STIX]{x1D70C}_{f}bL^{3}U_{0}^{2}/(EI)\in [7,81]$ (where $\unicode[STIX]{x1D70C}_{f}$ is the fluid density, $b$ and $L$ are the plate width and length, $U_{0}$ is the incoming velocity, $E$ is Young’s modulus and $I$ is the second moment of the area) and inclination angles $\unicode[STIX]{x1D6FC}$. Three-dimensional particle tracking velocimetry and a high-resolution force sensor were used to characterize the evolution of the plate dynamics and aerodynamic force. We show the existence of three distinctive, dominant modes of tip oscillations, which are modulated by the structure dynamic and flow instability. The first mode is characterized by small-amplitude, planar fluttering-like motions occurring under a critical Cauchy number, $Ca=Ca_{c}$. Past this condition, the motions are dominated by the second mode consisting of unsteady twisting superimposed onto the fluttering patterns. The onset of this mode is characterized by a sharp increase of the force fluctuation intensity. At sufficiently high $Ca$ and $\unicode[STIX]{x1D6FC}$, the plate may undergo a third mode given by large-scale tip orbits about the mean bending. Using the equation of motion and first-order approximations, we propose a formulation to estimate $Ca_{c}$ as a function of $\unicode[STIX]{x1D6FC}$; it exhibits solid agreement with experiments.

On the Dynamics of Flexible Plates under Rotational Motions

The reconfiguration of low-aspect-ratio flexible plates, required power and induced flow under pure rotation were experimentally inspected for various plate stiffness and angular velocities ω. Particle tracking velocimetry (PTV) and particle image velocimetry (PIV) were used to characterize the plate deformation along their span as well as the flow and turbulence statistics in the vicinity of the structures. Results show the characteristic role of stiffness and ω in modulating the structure reconfiguration, power required and induced flow. The inspected configurations allowed inspecting various plate deformations ranging from minor to extreme bending over 90∘ between the tangents of the two tips. Regardless of the case, the plates did not undergo noticeable deformation in the last ∼30% of the span. Location of the maximum deformation along the plate followed a trend s m ∝ l o g ( C a ) , where C a is the Cauchy number, which indicated that s m is roughly fixed at sufficiently large C a. The angle (α) between the plate in the vicinity of the tip and the tangential vector of the motions exhibited two distinctive, nearly-linear trends as a function of C a , within C a ∈ ( 0 , 15 ) and C a ∈ ( 20 , 70 ), with a matching within these C a at C a > 70, α ≈ 45 ∘. Induced flow revealed a local maximum of the turbulence levels at around 60% of the span of the plate; however, the largest turbulence enhancement occurred near the tip. Flexibility of the plate strongly modulated the spatial distribution of small-scale vortical structures; they were located along the plate wake in the stiffer plate and relatively concentrated near the tip in the low-stiffness plate. Due to relatively large deformation, rotational and wake effects, a simple formulation for predicting the mean reconfiguration showed offset; however, a bulk, constant factor on ω accounted for the offset between predictions and measurements at deformation reaching ∼ 60 ∘ between the tips.

Physical principles of optimization of the static regime of a cantilever type power-effect sensor with a constant rectangular cross-section

- In this paper an analysis of the physical principles of two-criterion optimization Pareto static mode of operation of power sensors cantilever type of rectangular type with a stable cross-section. The proposed criterion based on the Cauchy number is one of the characteristic numbers of the proportional miniaturization of microsystem technology. It is established that for a rectangular cantilever with a stable cross-section, the value of the Cauchy does not depend on the width of the micro-console and the material from which it is made.

Third and higher order convolution identities for Cauchy numbers

The n-th Cauchy number cn (n ? 0) are defined by the generating function x/ln(1+x)=??,n=0 cnxn=n!. In this paper, we deal with formulae of the type ?l1+...+lm=? l1,...,lm?0 ?!/l1!...lm! (cl1 +...+ clm )n = a0cn+? + ...+ am-1cn+?-m+1, where the ai are suitable rational numbers, the ci are Cauchy numbers and (cl1+...+clm )n := ?k1+...+km=n k1,...,km?0 n!/k1!...km! ck1+l1...ckm+lm. In particular, we give explicit formulae for m = 3 and m = 4.