Analysis of limit cycle oscillations of a typical airfoil section with freeplay

A typical airfoil section system with freeplay is investigated in the paper. The classic quasi-steady flow model is applied to calculate the aerodynamics, and a piecewise-stiffness model is adopted to characterize the nonlinearity of the airfoil section’s freeplay. There are two critical speeds in the system, i.e., a lower critical speed, above which the system might generate limit cycle oscillation, and an upper critical one, above which the system will flutter. Then a Poincaré map is constructed for the limit cycle oscillations by using piecewise-linear solutions with and without contact in the system. Through analysis of the Poincaré map, a series of equations which can determine the frequencies of period-1 limit cycle oscillations at any flight velocity are derived. Finally, these analytic results are compared to the results of numerical simulations, and a good agreement is found. The effects of freeplay value and contact stiffness ratio on the limit cycle oscillation are also analyzed through numerical simulations of the original system. Moreover, there exist multi-periods limit cycle oscillations and even complicated "chaotic" oscillations may occur, which are usually found in smooth nonlinear dynamic systems.     

A modified airfoil-based piezoaeroelastic energy harvester with double plunge degrees of freedom

In this letter,a piezoaeroelastic energy harvester based on an airfoil with double plunge degrees of freedom is proposed to additionally take advantage of the vibrational energy of the airfoil pitch motion.An analytical model of the proposed energy harvesting system is built and compared with an equivalent model using the well-explored pitch-plunge configuration.The dynamic response and average power output of the harvester are numerically studied as the flow velocity exceeds the cut-in speed(flutter speed).It is found that the harvester with double-plunge configuration generates 4%–10% more power with varying flow velocities while reducing 6% of the cut-in speed than its counterpart.     

Numerical investigation on a transition prediction model

A new transition prediction model is introduced,which couples the intermittency effect into the turbulence transport equations and takes the characteristics of fluid transition into consideration to mimic the exact process of transition.Test cases include a two-dimensional incompressible plate and a two-dimensional NACA0012 airfoil.Performance of this transition model for incompressible flows is studied,with numerical results consistent to experimental data.The requirement of grid resolution for this transition model is also studied.     

Numerical simulation of Gurney flap on SFYT15thick airfoil

A two-dimensional steady Reynolds-averaged Navier–Stokes(RANS) equation was solved to investigate the effects of a Gurney flap on SFYT15 thick airfoil aerodynamic performance. This airfoil was designed for flight vehicle operating at 20 km altitude with freestream velocity of 25 m/s. The chord length(C) is5 m and the Reynolds number based on chord length is Re = 7.76 × 105. Gurney flaps with the heights ranging from 0.25%C to 3%C were investigated. The shear stress transport(SST) k-ω turbulence model was used to simulate the flow structure around the airfoil. It is showed that Gurney flap can enhance not only the prestall lift but also lift-to-drag ratio in a certain range of angles of attack. Specially, at cruise angle of attack(α = 3°), Gurney flap with 0.5%C height can increase lift-to-drag ratio by 2.7%, and lift coefficient by 12.9%, respectively. Furthermore, the surface pressure distribution, streamlines and trailing-edge flow structure around the airfoil are illustrated, which are helpful to understand the mechanisms of Gurney flap on airfoil aerodynamic performance. Moreover, it is found that the increase of airfoil drag with Gurney flap can be attributed to the increase of pressure drag between the windward and the leeward sides of Gurney flap itself.     

Comparison Study of Constitutive Models for Overconsolidated Clays

Widely distributed in natural deposits,the overconsolidated(OC)clays have attracted extensive experimental investigations on their mechanical behaviors,especially in the 1960s and 1970s.Based on these results,numerous constitutive models have also been established.These models generally fall into two categories:one based on the classical plasticity theory and the other the bounding surface(BS)plasticity theory,with the latter being more popular and successful.The BS concept and the subloading surface(SS)concept are the two major BS plasticity theories.The features of these two concepts and the representative models based on them are introduced,respectively.The unified hardening(UH)model for OC clays is also based on the BS plasticity theory but distinguishes itself from other models by the integration of the reference yield surface,unified hardening parameter,potential failure stress ratio,arid transformed stress tensor.Modification is made to the Hvorslev envelop employed in the UH model to improve its capability of describing the behaviors of clays with extremely high overconsolidation ratio in this paper.The comparison among the BS model,SS model,and UH model is performed.Evidence shows that all these three models can characterize the fundamental behaviors of OC clays,such as the stress dilatancy,strain softening and attainment of the critical state.The UH model with the revised Hvorslev envelop has the fewest parameters which are identical to those of the modified Cam-Clay model.     

Transient analysis of diffusive chemical reactive species for couple stress fluid flow over vertical cylinder

The unsteady natural convective couple stress fluid flow over a semi-infinite vertical cylinder is analyzed for the homogeneous first-order chemical reaction effect. The couple stress fluid flow model introduces the length dependent effect based on the material constant and dynamic viscosity. Also, it introduces the biharmonic operator in the Navier-Stokes equations, which is absent in the case of Newtonian fluids. The solution to the time-dependent non-linear and coupled governing equations is carried out with an unconditionally stable Crank-Nicolson type of numerical schemes. Numerical results for the transient flow variables, the average wall shear stress, the Nusselt number, and the Sherwood number are shown graphically for both generative and destructive reactions. The time to reach the temporal maximum increases as the reaction constant K increases. The average values of the wall shear stress and the heat transfer rate decrease as K increases, while increase with the increase in the Sherwood number.     

A large eddy simulation of flows around an underwater vehicle model using an immersed boundary method

A large eddy simulation(LES) of the flows around an underwater vehicle model at intermediate Reynolds numbers is performed. The underwater vehicle model is taken as the DARPA SUBOFF with full appendages,where the Reynolds number based on the hull length is 1.0×105. An immersed boundary method based on the moving-least-squares reconstruction is used to handle the complex geometric boundaries. The adaptive mesh refinement is utilized to resolve the flows near the hull. The parallel scalabilities of the flow solver are tested on meshes with the number of cells varying from 50 million to 3.2 billion. The parallel solver reaches nearly linear scalability for the flows around the underwater vehicle model. The present simulation captures the essential features of the vortex structures near the hull and in the wake.Both of the time-averaged pressure coefficients and streamwise velocity profiles obtained from the LES are consistent with the characteristics of the flows pass an appended axisymmetric body. The code efficiency and its correct predictions on flow features allow us to perform the full-scale simulations on tens of thousands of cores with billions of grid points for higher-Reynolds-number flows around the underwater vehicles.     

Bifurcation analysis of fan casing under rotating air flow excitation

A fan casing model of cantilever circular thin shell is constructed based on the geometric characteristics of the thin-walled structure of aero-engine fan casing. According to Donnelly＇s shell theory and Hamilton＇s principle, the dynamic equations axe established. The dynamic behaviors are investigated by a multiple-scale method. The effects of casing geometric parameters and motion parameters on the natural frequency of the system are studied. The transition sets and bifurcation diagrams of the system are obtained through a singularity analysis of the bifurcation equation, showing that various modes of the system such as the bifurcation and hysteresis will appear in different parameter regions. In accordance with the multiple relationship of the fan speed and stator vibration frequency, the fan speed interval with the casing vibration sudden jump is calculated. The dynamic reasons of casing cracks are investigated. The possibility of casing cracking hysteresis interval is analyzed. The results show that cracking is more likely to appear in the hysteresis interval. The research of this paper provides a theoretical basis for fan casing design and system parameter optimization.     

Eigenvalue problem for integro-differential equation of supersonic panel,flutter

The dynamic stability of a thin plate in supersonic flow based on 2-dimensional linear theory leads to the study of a new problem in mathematical physics: complex eigenvalue prob-lem for a non-self-adjoint fourth-order integro-differential equation of Volterra’s type.Exact solutions of the aeroelastic system is obtained. In contrast to various approximate analyses, our critical curve agrees satisfactorily with experimental data, being free from divergence in the low supe’rsonic region. Moreover, we observe some notable physical behaviors: (1) mutual separation of flutter and vacuum frequency spectrums, (2) degeneracy of critical Mach number. The present method may be generalized in solving the supersonic flutter for 3-dimensional airfoil model as well as blade cascade in turbo-generator.     

EIGENVALUE PROBLEM FOR INTEGRO-DIFFERENTIAL EQUATION OF SUPERSONIC PANEL FLUTTER

The dynamic stability of a thin plate in supersonic flow based on 2-dimensional linear theory leads to the study of a new problem in mathematical physics: complex eigenvalue problem for a non-self-adjoint fourth-order integro-differential equation of Volterra's type. Exact solutions of the aeroelastic system is obtained. In contrast to various approximate analyses, our critical curve agrees satisfactorily with experimental data, being free from divergence in the low supersonic region. Moreover, we observe some notable physical behaviors: (1) mutual separation of flutter and vacuum frequency spectrums, (2) degeneracy of critical Mach number. The present method may be generalized in solving the supersonic flutter for 3-dimensional airfoil model as well as blade cascade in turbo-generator.     

Flow field characteristics study of a flapping airfoil using computational fluid dynamics

The flow field of a flapping airfoil in Low Reynolds Number (LRN) flow regime is associated with complex nonlinear vortex shedding and viscous phenomena. The respective fluid dynamics of such a flow is investigated here through Computational Fluid Dynamics (CFD) based on the Finite Volume Method (FVM). The governing equations are the unsteady, incompressible two-dimensional Navier-Stokes (N-S) equations. The airfoil is a thin ellipsoidal geometry performing a modified figure-of-eight-like flapping pattern. The flow field and vortical patterns around the airfoil are examined in detail, and the effects of several unsteady flow and system parameters on the flow characteristics are explored. The investigated parameters are the amplitude of pitching oscillations, phase angle between pitching and plunging motions, mean angle of attack, Reynolds number (Re), Strouhal number (St) based on the translational amplitudes of oscillations, and the pitching axis location (x/c). It is shown that these parameters change the instantaneous force coefficients quantitatively and qualitatively. It is also observed that the strength, interaction, and convection of the vortical structures surrounding the airfoil are significantly affected by the variations of these parameters.     

High-fidelity simulations of moving and flexible airfoils at low Reynolds numbers

The present paper highlights results derived from the application of a high-fidelity simulation technique to the analysis of low-Reynolds-number transitional flows over moving and flexible canonical configurations motivated by small natural and man-made flyers. This effort addresses three separate fluid dynamic phenomena relevant to small fliers, including: laminar separation and transition over a stationary airfoil, transition effects on the dynamic stall vortex generated by a plunging airfoil, and the effect of flexibility on the flow structure above a membrane airfoil. The specific cases were also selected to permit comparison with available experimental measurements. First, the process of transition on a stationary SD7003 airfoil section over a range of Reynolds numbers and angles of attack is considered. Prior to stall, the flow exhibits a separated shear layer which rolls up into spanwise vortices. These vortices subsequently undergo spanwise instabilities, and ultimately breakdown into fine-scale turbulent structures as the boundary layer reattaches to the airfoil surface. In a time-averaged sense, the flow displays a closed laminar separation bubble which moves upstream and contracts in size with increasing angle of attack for a fixed Reynolds number. For a fixed angle of attack, as the Reynolds number decreases, the laminar separation bubble grows in vertical extent producing a significant increase in drag. For the lowest Reynolds number considered (Re _c  = 10⁴), transition does not occur over the airfoil at moderate angles of attack prior to stall. Next, the impact of a prescribed high-frequency small-amplitude plunging motion on the transitional flow over the SD7003 airfoil is investigated. The motion-induced high angle of attack results in unsteady separation in the leading edge and in the formation of dynamic-stall-like vortices which convect downstream close to the airfoil. At the lowest value of Reynolds number (Re _c  = 10⁴), transition effects are observed to be minor and the dynamic stall vortex system remains fairly coherent. For Re _c  = 4 × 10⁴, the dynamic-stall vortex system is laminar at is inception, however shortly afterwards, it experiences an abrupt breakdown associated with the onset of spanwise instability effects. The computed phased-averaged structures for both values of Reynolds number are found to be in good agreement with the experimental data. Finally, the effect of structural compliance on the unsteady flow past a membrane airfoil is investigated. The membrane deformation results in mean camber and large fluctuations which improve aerodynamic performance. Larger values of lift and a delay in stall are achieved relative to a rigid airfoil configuration. For Re _c = 4.85 × 10⁴, it is shown that correct prediction of the transitional process is critical to capturing the proper membrane structural response.     

基于涡粘性模型的翼型湍流流场熵产率计算

利用有限体积法实现了基于非正交同位网格的SIMPLE算法。基于熵分析方法，采用涡粘性模型求解湍流熵产方程，系统研究了湍流模型对二维翼型绕流流场熵产率的影响。通过计算NACA0012翼型在来流雷诺数为2.88×10⁶时，0°攻角~16.5°攻角范围内的翼型表面压力系数分布和升阻力特性，验证了算法及程序的正确性。结果表明，选择不同湍流模型时，翼型流场熵产的计算结果存在差异，湍流耗散是引起流场熵产的主要原因；翼型流场的熵产主要发生在翼型前缘区、壁面边界层和翼型尾流区域，流场熵产率与翼型阻力系数线性相关；当产生分离涡时，粘性耗散引起的熵产下降。     

Buffeting in transonic flow prediction using time‐dependent turbulence model

In transonic flow conditions, the shock wave/turbulent boundary layer interaction and flow separations on wing upper surface induce flow instabilities, ‘buffet’, and then the buffeting (structure vibrations). This phenomenon can greatly influence the aerodynamic performance. These flow excitations are self‐sustained and lead to a surface effort due to pressure fluctuations. They can produce enough energy to excite the structure. The objective of the present work is to predict this unsteady phenomenon correctly by using unsteady Navier–Stokes‐averaged equations with a time‐dependent turbulence model based on the suitable (k–ε) turbulent eddy viscosity model. The model used is based on the turbulent viscosity concept where the turbulent viscosity coefficient (Cμ) is related to local deformation and rotation rates. To validate this model, flow over a flat plate at Mach number of 0.6 is first computed, then the flow around a NACA0012 airfoil. The comparison with the analytical and experimental results shows a good agreement. The ONERA OAT15A transonic airfoil was chosen to describe buffeting phenomena. Numerical simulations are done by using a Navier–Stokes SUPG (streamline upwind Petrov–Galerkin) finite‐element solver. Computational results show the ability of the present model to predict physical phenomena of the flow oscillations. The unsteady shock wave/boundary layer interaction is described. Copyright © 2005 John Wiley & Sons, Ltd.     

Feedback control of laminar flow separation on NACA23012 airfoil by POD analysis and using perturbed Navier‐Stokes equations

The main purpose of this article is to develop a forced reduced‐order model based on the proper orthogonal decomposition (POD)/Galerkin projection (on isentropic Navier‐Stokes equations) and perturbation method on the compressible Navier‐Stokes equations. The resulting forced reduced‐order model will be used in optimal control of the separated flow over a NACA23012 airfoil at Mach number of 0.2, Reynolds number of 800, and high incidence angle of 24°. The main disadvantage of the POD/Galerkin projection method for control purposes is that controlling parameters do not show up explicitly in the resulting reduced‐order system. The perturbation method and POD/Galerkin projection on the isentropic Navier‐Stokes equations introduce a forced reduced‐order model that can predict the time varying influence of the controlling parameters and the Navier‐Stokes response to external excitations. An optimal control theory based on forced reduced‐order system is used to design a control law for a nonlinear reduced‐order system, which attempts to minimize the vorticity content in the flow field. The test bed is a laminar flow over NACA23012 airfoil actuated by a suction jet at 12% to 18% chord from leading edge and a pair of blowing/suction jets at 15% to 18% and 24% to 30% chord from leading edge, respectively. The results show that wall jet can significantly influence the flow field, remove separation bubbles, and increase the lift coefficient up to 22%, while the perturbation method can predict the flow field in an accurate manner.     

A method to carry out shape optimization with a large number of design variables

A new method for shape optimization with relatively large number of design variables is proposed. It is well known that gradient‐based methods converge to a local optimum. As a result, utilization of a richer design space does not necessarily lead to a better design. This is demonstrated via the design of an airfoil for maximum lift for Re = 1000 and α = 4° flow. The airfoil is represented by fourth‐order non‐uniform rational B‐splines, and the control points are used as design variables. Starting with a NACA0012 airfoil, it is found that the optimal airfoil obtained with 13 control points has far superior aerodynamic performance than the ones obtained with 39 and 61 control points. For effective utilization of a richer design space, it is proposed that the number of design variables be increased gradually. The method is demonstrated by designing high lift airfoils for Re = 1000 and 1 × 10⁴. The objective function is the maximization of the time‐averaged lift coefficient for α = 4°. The optimization cycle with 27 control points is initiated with the optimal airfoil obtained with 13 control points. The process is continued with gradual increase in the number of design variables. Beyond a certain number of control points, the optimization leads to a spontaneous appearance of corrugations on the upper surface of the airfoil. The corrugations are responsible for the generation of small vortices that add to the suction on the upper surface of the airfoil and lead to enhanced lift. A stabilized finite element method is used to solve the unsteady flow and adjoint equations. Copyright © 2012 John Wiley & Sons, Ltd.     

Input-output reduced-order modeling of unsteady flow over an airfoil at a high angle of attack based on dynamic mode decomposition with control

Due to the damage caused by stall flutter, the investigation and modeling of the flow over a wind turbine airfoil at high angles of attack are essential. Dynamic mode decomposition (DMD) and dynamic mode decomposition with control (DMDc) are used to analyze unsteady flow and identify the intrinsic dynamics. The DMDc algorithm is found to have an identification problem when the spatial dimension of the training data is larger than the number of snapshots. IDMDc, a variant algorithm based on reduced dimension data, is introduced to identify the precise intrinsic dynamics. DMD, DMDc and IDMDc are all used to decompose the data for unsteady flow over the S809 airfoil that are obtained by numerical simulations. The DMD results show that the dominant feature of a static airfoil is the adjacent shedding vortices in the wake. For an oscillating airfoil, the DMDc results may fail to consider the effect of the input and have an identification problem. IDMDc can alleviate this problem. The dominant IDMDc modes show that the intrinsic flow for the oscillating case is similar to the unsteady flow over the static airfoil. Moreover, the input–output model identified by IDMDc can give better predictions for different oscillating cases than the identified DMDc model.     

Direct numerical simulation of a NACA0012 in full stall

This work aims at investigating the mechanisms of separation and the transition to turbulence in the separated shear-layer of aerodynamic profiles, while at the same time to gain insight into coherent structures formed in the separated zone at low-to-moderate Reynolds numbers. To do this, direct numerical simulations of the flow past a NACA0012 airfoil at Reynolds numbers Re = 50,000 (based on the free-stream velocity and the airfoil chord) and angles of attack AOA = 9.25° and AOA = 12° have been carried out. At low-to-moderate Reynolds numbers, NACA0012 exhibits a combination of leading-edge/trailing-edge stall which causes the massive separation of the flow on the suction side of the airfoil. The initially laminar shear layer undergoes transition to turbulence and vortices formed are shed forming a von Kármán like vortex street in the airfoil wake. The main characteristics of this flow together with its main features, including power spectra of a set of selected monitoring probes at different positions on the suction side and in the wake of the airfoil are provided and discussed in detail.     

Experimental study of vortex–structure interaction noise radiated from rod–airfoil configurations

Vortex–structure interaction noise radiated from an airfoil embedded in the wake of a rod is investigated experimentally in an anechoic wind tunnel by means of a phased microphone array for acoustic tests and particle image velocimetry (PIV) for the flow field measurements. The rod–airfoil configuration is varied by changing the rod diameter (D), adjusting the cross-stream position (Y) of the rod and the streamwise gap (L) between the rod and the airfoil leading edge. Two noise control concepts, including “air blowing” on the upstream rod and a soft-vane leading edge on the airfoil, are applied to control the vortex–structure interaction noise. The motivation behind this study is to investigate the effects of the three parameters on the characteristics of the radiated noise and then explore the influences of the noise control concepts. Both the vortex–structure interaction noise and the rod vortex shedding tonal noise are analysed. The acoustic test results show that both the position and magnitude of the dominant noise source of the rod–airfoil model are highly dependent on the parameters considered. In the case where the vortex–structure interaction noise is dominant, the application of the air blowing and the soft vane can effectively attenuate the interaction noise. Flow field measurements suggest that the intensity of the vortex–structure interaction and the flow impingement on the airfoil leading edge are suppressed by the control methods, giving a reduction in noise.     

Exploration of plasma-based control for low-Reynolds number airfoil/gust interaction

Large-eddy simulation (LES) is employed to investigate the use of plasma-based actuation for the control of a vortical gust interacting with a wing section at a low Reynolds number. Flow about the SD7003 airfoil section at 4° angle of attack and a chord-based Reynolds number of 60,000 is considered in the simulation, which typifies micro air vehicle (MAV) applications. Solutions are obtained to the Navier–Stokes equations that were augmented by source terms used to represent body forces imparted by the plasma actuator on the fluid. A simple phenomenological model provided these body forces resulting from the electric field generated by the plasma. The numerical method is based upon a high-fidelity time-implicit scheme and an implicit LES approach which are used to obtain solutions on a locally refined overset mesh system. A Taylor-like vortex model is employed to represent a gust impinging upon the wing surface, which causes a substantial disruption to the undisturbed flow. It is shown that the fundamental impact of the gust on unsteady aerodynamic forces is due to an inviscid process, corresponding to variation in the effective angle of attack, which is not easily overcome. Plasma control is utilised to mitigate adverse effects of the interaction and improve aerodynamic performance. Physical characteristics of the interaction are described, and several aspects of the control strategy are explored. Among these are uniform and non-uniform spanwise variations of the control configuration, co-flow and counter-flow orientations of the directed force, pulsed and continuous operations of the actuator and strength of the plasma field. Results of the control situations are compared with regard to their effect upon aerodynamic forces. It was found that disturbances to the moment coefficient produced by the gust can be greatly reduced, which may be significant for stability and handling of MAV operations.