Reduced coupled flapping wing-fluid computational model with unsteady vortex wake

Insect flight research is propelled by their unmatched flight capabilities. However, complex underlying aerodynamic phenomena make computational modeling of insect-type flapping flight a challenging task, limiting our ability in understanding insect flight and producing aerial vehicles exploiting same aerodynamic phenomena. To this end, novel mid-fidelity approach to modeling insect-type flapping vehicles is proposed. The approach is computationally efficient enough to be used within optimal design and optimal control loops, while not requiring experimental data for fitting model parameters, as opposed to widely used quasi-steady aerodynamic models. The proposed algorithm is based on Helmholtz–Hodge decomposition of fluid velocity into curl-free and divergence-free parts. Curl-free flow is used to accurately model added inertia effects (in almost exact manner), while expressing system dynamics by using wing variables only, after employing symplectic reduction of the coupled wing-fluid system at zero level of vorticity (thus reducing out fluid variables in the process). To this end, all terms in the coupled body-fluid system equations of motion are taken into account, including often neglected terms related to the changing nature of the added inertia matrix (opposed to the constant nature of rigid body mass and inertia matrix). On the other hand—in order to model flapping wing system vorticity effects—divergence-free part of the flow is modeled by a wake of point vortices shed from both leading (characteristic for insect flight) and trailing wing edges. The approach is evaluated for a numerical case involving fruit fly hovering, while quasi-steady aerodynamic model is used as benchmark tool with experimentally validated parameters for the selected test case. The results indicate that the proposed approach is capable of mid-fidelity accurate calculation of aerodynamic loads on the insect-type flapping wings.

     

昆虫拍翼方式的非定常流动物理再探讨

基于提出的理论模化方法来探讨昆虫拍翼方式的非定常流动物理. 以悬停飞行为 例,通过对拍翼运动的分析,不仅解释了昆虫利用高频拍翼的方式为何能够克服低雷诺数带 来的气动局限性(St \gg 1/Re),而且还指出高升力产生和调节的3个流动 控制因素：(1) 由于拍翼的变速运动即时引起了流体动力响应,这种附加惯性效应 可产生瞬时的高升力; (2) 保持前缘涡不脱离翼面有助于减少升力的下降; (3) 增大后缘涡的强度并加速其脱离后缘能够有效地提高升力.     

基于三路舵机驱动的扑翼飞行模型气动特性研究

扑动而形成非定常气动现象是扑翼飞行过程中产生高升力的主要原因。本文以Ellington实验的鹰蛾翅膀为原形,设计扑翼实验及数值计算模型。通过压差传感器对翅膀模型上翼面固定位置进行测压,分析前缘涡的产生及脱落情况(考虑动压效应)。测量上下翼面固定位置处的压差,揭示扑翼飞行中产生高升力的主要原因。利用烟风洞观察扑翼模型周围流场结构及特殊涡产生变化情况。另外,根据Ellington提供的升力关系式估算了扑翼模型在一个周期内的平均升力。最后,基于三维欧拉方程对扑翼飞行气动特性进行数值模拟,计算结果与实验吻合良好。     

Design and development considerations for biologically inspired flapping-wing micro air vehicles

In this paper, the decade of numerical and experimental investigations leading to the development of the authors’ unique flapping-wing micro air vehicle is summarized. Early investigations included the study of boundary layer energization by means of a small flapping foil embedded in a flat-plate boundary layer, the reduction of the recirculatory flow region behind a backward-facing step by means of a small flapping foil, and the reduction or suppression of flow separation behind blunt or cusped airfoil trailing edges by flapping a small foil located in the wake flow region. These studies were followed by systematic investigations of the aerodynamic characteristics of single flapping airfoils and airfoil combinations. These unsteady flows were described using flow visualization, laser-Doppler velocimetry in addition to panel and Navier–Stokes computations. It is then shown how this flapping-wing database was used to conceive, design and develop a micro air vehicle which has a fixed wing for lift and two flapping wings for thrust generation. While animal flight is characterized by a coupled force generation, the present design appears to separate lift and thrust. However, in fact, the performance of one surface is closely coupled to the other surfaces.     

Aerodynamic characteristics of flapping motion in hover

The aim of the present work is to understand the aerodynamic phenomena and the vortex topology of an unsteady flapping motion by means of numerical and experimental methods. Instead of the use of real insect/bird wing geometries and kinematics which are highly complex and difficult to imitate by an exact modeling, a simplified model is used in order to understand the unsteady aerodynamics and vortex formation mechanisms during the different phases of the flapping motion. The flow is assumed to be laminar with a Reynolds number of 1,000. Direct numerical simulations, laser sheet visualizations and particle image velocimetry (PIV) measurements are performed for the phenomenological analysis of the flow. The vortex dynamics and their identification are put in evidence with PIV measurements by considering velocity magnitude, streamlines, second invariant of velocity gradient (Q-criteria), vorticity contours and Eurlerian accelerations.     

Using stereo multigrid DPIV (SMDPIV) measurements to investigate the vortical skeleton behind a finite-span flapping wing

The structure of the flow behind wings with finite span (3D) is significantly more complex than the flow behind infinite span (2D) wings. It has been shown that the presence of wingtip vortices behind finite span wings significantly modifies the geometry of the wake flow. It is felt that this modification alters the dynamics of interaction between leading and trailing edge vorticity in a manner that affects the ability of 2D flapping wings to produce thrust. A model of the mean flow skeleton has been proposed from qualitative flow visualization experiments. An unambiguous quantitative representation of the actual flow is required for comparison to the proposed model. To accomplish this the full 3D 3C velocity is required in the volume behind the 3D flapping wing. It is proposed to use stereoscopic multigrid digital particle image velocimetry (SMDPIV) measurements to investigate this unsteady oscillatory flow. This paper reports preliminary SMDPIV measurements along the plane of a symmetrical NACA-profile wing at a Strouhal number of 0.35. Phase averaged measurements are used to investigate the complex flow topology and the influence of the forcing flow on the evolution of the large scale structure of a jet-flow. This paper focuses on optimizing the SMDPIV experimental methodology applied to liquid flows. By refining the 2D 3C technique, the 3D topology of the flow can be investigated with a high degree of accuracy and repeatability. Preliminary results show that the flow is characterized by two pairs of coherent structures of positive and negative vorticity. The arrangement of these structures in the flow is controlled by the motion of the wing. Vorticity of opposite rotation is shed at the extreme heave and pitch positions of the aerofoil to set up a thrust indicative vortex street in support of the suggested topological model.     

An experimental study of the unsteady vortex structures in the wake of a root-fixed flapping wing

An experimental study was conducted to characterize the evolution of the unsteady vortex structures in the wake of a root-fixed flapping wing with the wing size, stroke amplitude, and flapping frequency within the range of insect characteristics for the development of novel insect-sized nano-air-vehicles (NAVs). The experiments were conducted in a low-speed wing tunnel with a miniaturized piezoelectric wing (i.e., chord length, C = 12.7 mm) flapping at a frequency of 60 Hz (i.e., f = 60 Hz). The non-dimensional parameters of the flapping wing are chord Reynolds number of Re = 1,200, reduced frequency of k = 3.5, and non-dimensional flapping amplitude at wingtip h = A/C = 1.35. The corresponding Strouhal number (Str) is 0.33, which is well within the optimal range of 0.2 < Str < 0.4 used by flying insects and birds and swimming fishes for locomotion. A digital particle image velocimetry (PIV) system was used to achieve phased-locked and time-averaged flow field measurements to quantify the transient behavior of the wake vortices in relation to the positions of the flapping wing during the upstroke and down stroke flapping cycles. The characteristics of the wake vortex structures in the chordwise cross planes at different wingspan locations were compared quantitatively to elucidate underlying physics for a better understanding of the unsteady aerodynamics of flapping flight and to explore/optimize design paradigms for the development of novel insect-sized, flapping-wing-based NAVs.     

Nonlinear characteristics of a thin wing in a shock-free flow around the nose

An effective means of controlling wing leading-edge stall at high angles of attack is deflection of the nose in order to assure shock-free entrance of the stream. A numerical method of computing the angles of nose deflection and the aerodynamic characteristics of a thin wing of arbitrary planform for a shock-free entrance of the steady ideal incompressible fluid stream is elucidated in this paper on the basis of nonlinear wing theory [1]. The problem is solved by the method of discrete vortices. In the computations, the wing and its wake, replaced by a vortex sheet, are modeled by a system of discrete vortices which are nonlinear segments with constant circulation along the length. The angles of deflection of the nose and the aerodynamic characteristics of the wing, including shunting of the free vortices shed from the side and trailing edges, are determined during the computation. Examples of an electronic digital computer are presented.     

Mathematical simulation of unsteady flow past a wing with jets

The various approximate approaches to the investigation of the unsteady aerodynamic characteristics of an airfoil with jet flap [1–3] are applicable only for an airfoil, low jet intensity, and low oscillation frequencies. In the present paper, the method of discrete vortices [4] is generalized to the case of unsteady flow past a wing with jets and arbitrary shape in plan. The problem is solved in the linear formulation; the conditions used are standard: no flow through the wing and jet, finite velocities at the trailing edges where there is no jet, and also a dynamical condition on the jet. The wing and jet are assumed to be thin and the medium inviscid and incompressible.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 139–144, May–June, 1982.     

Reynolds number effects on leading edge vortex development on a waving wing

The waving wing experiment is a fully three-dimensional simplification of the flapping wing motion observed in nature. The spanwise velocity gradient and wing starting and stopping acceleration that exist on an insect-like flapping wing are generated by rotational motion of a finite span wing. The flow development around a waving wing at Reynolds number between 10,000 and 60,000 has been studied using flow visualization and high-speed PIV to capture the unsteady velocity field. Lift and drag forces have been measured over a range of angles of attack, and the lift curve shape was similar in all cases. A transient high-lift peak approximately 1.5 times the quasi-steady value occurred in the first chord length of travel, caused by the formation of a strong attached leading edge vortex. This vortex appears to develop and shed more quickly at lower Reynolds numbers. The circulation of the leading edge vortex has been measured and agrees well with force data.     

Calculation of flow around a lifting surface in the case of large deformations

Many studies have been made of the nonstationary flow of an ideal incompressible fluid around a lifting surface. The present state of the numerical methods of solution of this problem is reviewed in [1]. The present paper studies three-dimensional nonstationary flow around a lifting surface which undergoes deformation and behind which a wake vortex surface is formed. The lifting and wake vortex surfaces are represented in parametric form. The metrics of these surfaces are used, and the introduced vortex function is approximated by bicubic splines. For the convenient application of the theory developed here to the flapping flight of insects, for which it is sometimes difficult to distinguish the lateral and trailing edges of the wings, the following terminology is introduced. The part of the edge of the lifting surface from which the wake vortex surface is shed is called the trailing edge. The remaining part is called the leading edge. On the leading edge, the velocity has a singularity. Test calculations have demonstrated the effectiveness of the method.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 72–79, July–August, 1980.     

Unsteady flow calculation in a radial flow centrifugal pump with spiral casing

The prediction of the two-dimensional unsteady flow established in a radial flow centrifugal pump is considered. Assuming the fluid incompressible and inviscid, the velocity field is represented by means of source and vorticity surface distributions as well as a set of point vortices. Using this representation, a grid-free (Lagrangian) numerical method is derived based on the coupling of the boundary element and vortex particle methods. In this context the source and vorticity surface distributions are determined through the non-entry boundary condition together with the unsteady Kutta condition. In order to satisfy Kelvin's theorem, vorticity is shed at the trailing edges of the impeller blades. Then the vortex particle method is used to approximate the convection of the free vorticity distribution. Results are given for a pump configuration experimentally tested by Centre Technique des Industries Mécaniques (CETIM). Comparisons between predictions and experimental data show the capability of the proposed method to reproduce the main features of the flow considered.     

Numerical investigation on aerodynamic performance of a bionic flapping wing

This paper numerically studies the aerodynamic performance of a bird-like bionic flapping wing. The geometry and kinematics are designed based on a seagull wing,in which flapping, folding, swaying, and twisting are considered. An in-house unsteady flow solver based on hybrid moving grids is adopted for unsteady flow simulations. We focus on two main issues in this study, i.e., the influence of the proportion of down-stroke and the effect of span-wise twisting. Numerical results show that the proportion of downstroke is closely related to the efficiency of the flapping process. The preferable proportion is about 0.7 by using the present geometry and kinematic model, which is very close to the observed data. Another finding is that the drag and the power consumption can be greatly reduced by the proper span-wise twisting. Two cases with different reduced frequencies are simulated and compared with each other. The numerical results show that the power consumption reduces by more than 20%, and the drag coefficient reduces by more than 60% through a proper twisting motion for both cases. The flow mechanism is mainly due to controlling of unsteady flow separation by adjusting the local effective angle of attack. These conclusions will be helpful for the high-performance micro air vehicle(MAV) design.     

Estimation of the losses in hydrodynamic cascades due to periodic flow unsteadiness

The losses in hydrodynamic cascades caused by periodic flow unsteadiness are theoretically estimated on the assumption that the losses are due to energy expenditure to the formation of unsteady trailing vortices shed from the profiles of the cascades. The trailing vortex intensity is determined within the framework of the inviscid fluid model, by solving the corresponding problems in the linear formulation. The work done on the trailing vortex formation is determined by the increment of the kinetic energy of the flow induced by the corresponding vortices. Examples of calculations are presented for the case of periodic unsteadiness of the flow due to the hydrodynamic interaction of the cascades. The calculated results are compared with the experimental data.     

Mathematical modeling of the sound induced in a subsonic flow around cascades in relative motion

A method of mathematical modeling of the tonal sound induced by the unsteady aerodynamic interaction of two plane airfoil cascades in a subsonic flow and in uniform relative motion in the direction of their fronts is developed. The method is based on the numerical integration of the unsteady flow equations using a simplified model for the periodic system of edge wakes shed from the airfoils of the first (leading) cascade in the viscous flow and acting on the second (trailing) cascade. An analysis of the distinctive features of the flow under consideration demonstrates the efficiency of the model proposed.     

NUMERICAL SIMULATION OF INSECT FLIGHT

In the non-inertial coordinates attached to the model wing, the two-dimensional unsteady flow field triggered by the motion of the model wing, similar to the flapping of the insect wings, was numerically simulated. One of the advantages of our method is that it has avoided the difficulty related to the moving-boundary problem. Another advantage is that the model has three degrees of freedom and can be used to simulate arbitrary motions of a two-dimensional wing in plane only if the motion is known. Such flexibility allows us to study how insects control their flying. Our results show that there are two parameters that are possibly utilized by insects to control their flight: the phase difference between the wing translation and rotation, and the lateral amplitude of flapping along the direction perpendicular to the average flapping plane.     

Stereo PIV measurement of a finite, flapping rigid plate in hovering condition

Qualitative and quantitative flow visualizations were performed on a flapping rigid plate to establish a quantitative method for flow observation and evaluation of the force in the near field of a flapping wing. Flow visualization was performed qualitatively with dye visualization and quantitatively with velocity measurements using stereo particle image velocimetry (PIV) on three planes near the tip of the plate along its chord and oriented normally. By ensemble averaging the velocity fields of the same phase angles, they represent a portion of the volume near the tip. Measurements were conducted with two flapping frequencies to compare the flow structure. The second invariant of the deformation tensor visualized the leading edge and mid-chord vortices around the plate appearing due to flow separation behind the plate while other vortical structures were visualized by streamlines. These structures appear to be related to the dynamics of the leading edge vortex. Force analysis by integrating the phase-averaged velocity field within a chosen control volume showed increases in the maxima of the magnitudes of the non-dimensional unsteady force terms on the edge of the plate at the angles after the end of each stroke. The non-dimensional phase-averaged momentum flux was similar for both flapping frequencies.     

Aerodynamic mechanism of forces generated by twisting model-wing in bat flapping flight

The aerodynamic mechanism of the bat wing membrane Mong the lateral border of its body is studied. The twist-morphing that alters the angle of attack （AOA） along the span-wise direction is observed widely during bat flapping flight. An assumption is made that the linearly distributed AOA is along the span-wise direction. The plate with the aspect ratio of 3 is used to model a bat wing. A three-dimensional （3D） unsteady panel method is used to predict the aerodynamic forces generated by the flapping plate with leading edge separation. It is found that, relative to the rigid wing flapping, twisting motion can increase the averaged lift by as much as 25% and produce thrust instead of drag. Furthermore, the aerodynamic forces （lift/drag） generated by a twisting plate-wing are similar to those of a pitching rigid-wing, meaning that the twisting in bat flight has the same function as the supination/pronation motion in insect flight.     

High fidelity numerical simulation of airfoil thickness and kinematics effects on flapping airfoil propulsion

High-fidelity numerical simulations with the spectral difference (SD) method are carried out to investigate the unsteady flow over a series of oscillating NACA 4-digit airfoils. Airfoil thickness and kinematics effects on the flapping airfoil propulsion are highlighted. It is confirmed that the aerodynamic performance of airfoils with different thickness can be very different under the same kinematics. Distinct evolutionary patterns of vortical structures are analyzed to unveil the underlying flow physics behind the diverse flow phenomena associated with different airfoil thickness and kinematics and reveal the synthetic effects of airfoil thickness and kinematics on the propulsive performance. Thickness effects at various reduced frequencies and Strouhal numbers for the same chord length based Reynolds number (=1200) are then discussed in detail. It is found that at relatively small Strouhal number (=0.3), for all types of airfoils with the combined pitching and plunging motion (pitch angle 20°, the pitch axis located at one third of chord length from the leading edge, pitch leading plunge by 75°), low reduced frequency (=1) is conducive for both the thrust production and propulsive efficiency. Moreover, relatively thin airfoils (e.g. NACA0006) can generate larger thrust and maintain higher propulsive efficiency than thick airfoils (e.g. NACA0030). However, with the same kinematics but at relatively large Strouhal number (=0.45), it is found that airfoils with different thickness exhibit diverse trend on thrust production and propulsive efficiency, especially at large reduced frequency (=3.5). Results on effects of airfoil thickness based Reynolds numbers indicate that relative thin airfoils show superior propulsion performance in the tested Reynolds number range. The evolution of leading edge vortices and the interaction between the leading and trailing edge vortices play key roles in flapping airfoil propulsive performance.     

Asymptotic theory of flow separation from wings of low aspect ratio

Numerous methods have been developed to calculate the aerodynamic characteristics of wings of low aspect ratio in the case when there is flow separation from the wing edges. Among the methods based on direct solution of the three-dimensional Euler equations there are the method of discrete vortices [1, 2] and the panel method [3]. In addition, numerical and asymptotic methods [4, 5] based on the theory of slender bodies [6] are used. One of the most important shortcomings of this theory is the dependence of the flow pattern at a given section of the wing on only the upstream flow. The obtained solutions therefore contain no information about the influence of the trailing edge of the wing, on which, as is well known, the Chaplygin-Zhukovskii condition is satisfied. The aim of the present paper is to construct an asymptotic theory of higher approximation and a corresponding numerical method for calculating flow separation from wings of low aspect ratio in which this shortcoming is absent.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 141–147, July–August, 1982.