Numerical investigation of transonic limit cycle oscillations of a two-dimensional supercritical wing

CFD-based aeroelastic computations are performed to investigate the effect of nonlinear aerodynamics on transonic limit cycle oscillation (LCO) characteristics of the NLR7301 airfoil section. It is found that the LCO solutions from Navier–Stokes computations deviate less from the experiment than an Euler solution but strongly depend on the employed turbulence model. The Degani–Schiff modification to the Baldwin–Lomax turbulence model provokes spurious vorticity spots causing multiple shocks which might be unphysical, while the Spalart–Allmaras turbulence model yields physically reasonable unsteady shocks. In the cases examined, smaller initial perturbations lead to larger LCO amplitudes and vice versa, in contradiction to what one might expect. The amplitude of the initial perturbation is also found to have an impact on the mean position of LCO. Also addressed in the paper are aspects of multiblock message passing interface (MPI) parallel computation techniques as related to the present problem.     

用隐式方法和WENO格式计算悬停旋翼跨声速无粘流场

寻找一种能够准确计算以涡为主要特征的复杂流场和克服尾迹耗散问题的数值方法,一直是旋翼空气动力学研究的热点和难点。本文发展了一种基于高阶迎风格式计算悬停旋翼无粘流场的隐式数值方法。无粘通量采用Roe通量差分分裂格式,为提高精度,使用五阶WENO格式进行左右状态插值,并与MUSCL插值进行比较。为提高收敛到定常解的效率,时间推进采用LU-SGS隐式方法。用该方法对一跨声速悬停旋翼无粘流场进行了数值计算,数值结果表明WENO-Roe的激波分辨率高于MUSCL-Roe,体现出了格式精度的提高对计算结果的改善,LU-SGS隐式方法的计算效率比5步Runge-Kutta显式方法的高。     

Power extraction from aeroelastic limit cycle oscillations

Nonlinear limit cycle oscillations of an aeroelastic energy harvester are exploited for enhanced piezoelectric power generation from aerodynamic flows. Specifically, a flexible beam with piezoelectric laminates is excited by a uniform axial flow field in a manner analogous to a flapping flag such that the system delivers power to an electrical impedance load. Fluid–structure interaction is modeled by augmenting a system of nonlinear equations for an electroelastic beam with a discretized vortex-lattice potential flow model. Experimental results from a prototype aeroelastic energy harvester are also presented. Root mean square electrical power on the order of 2.5 mW was delivered below the flutter boundary of the test apparatus at a comparatively low wind speed of 27 m/s and a chord normalized limit cycle amplitude of 0.33. Moreover, subcritical limit cycles with chord normalized amplitudes of up to 0.46 were observed. Calculations indicate that the system tested here was able to access over 17% of the flow energy to which it was exposed. Methods for designing aeroelastic energy harvesters by exploiting nonlinear aeroelastic phenomena and potential improvements to existing relevant aerodynamic models are also discussed.     

Numerical simulation of flow-induced bi-directional oscillations

Flow-induced vibration (FIV) by vortex shedding behind a submerged cylinder can lead to damage of nuclear components. With respect to such a serious scenario, various experiments and numerical simulations have been conducted to predict the vibration phenomena. Especially in simulation, the immersed finite element method (IFEM) is a promising approach to solve fluid-structure interaction problems because it needs less computational resources. In this paper, two-dimensional motions of cylinders are simulated by using IFEM to obtain their vibration characteristics. Three benchmark tests such as flow past a fixed circular cylinder, in-line oscillation of a circular cylinder and flow-induced vibration with uni-directional motion are performed to verify the proposed numerical method. Furthermore, bi-directional motions of two horizontally and vertically arranged cylinders as well as that of a single cylinder in fluid flow are analyzed, and then key findings are fully discussed.     

高精度PPM格式在湍流燃烧中的应用

利用处理三维可压缩粘性流体流动问题中的沉浸边界法,并结合基于PPM方法的高精度TVD格式,对三维方形管道中部的圆柱火焰绕流及惰性气体绕流问题进行了数值模拟。计算湍流时采用大涡模拟(LES),化学反应速率采用EBU漩涡破碎模型。通过计算结果与实验结果的比较,发现高精度PPM格式能精确模拟两类圆柱绕流问题。计算中还发现,火焰圆柱绕流算例中,在火焰翻越圆柱前,由于燃烧的膨胀作用,使得火焰正面前的未燃气体流动并形成惰性气体绕流,这与无燃烧时的惰性气体绕流类似。但当火焰翻越圆柱过程中及完全翻越圆柱后,两种算例绕流流场出现明显变化。     

Numerical simulation of viscous transonic flows over an airfoil

High Reynolds number viscous transonic flow is described based on an interaction of the potential outer flow with the boundary layer and wake. Following the procedure of Lighthill (1958), the solutions in these domains are matched to each other through boundary conditions. The solution to the complete problem is obtained iteratively through successive computations of the flows in the outer and inner domains. Both old and new algorithms are used for the iteration process and subsequent problem solution. Results are given for all the airfoils from the Experimental Data Base for Computer Program Assessment (AGARD-AR-138, 1979). A comparison of these results with experimental data shows the degree of agreement between these unbounded airfoil flow simulations and real transonic flow over the central part of a straight wing.     

Flutter control and mitigation of limit cycle oscillations in aircraft wings using distributed vibration absorbers

Nonlinear Dynamics - In this work, we demonstrate the application of the conserved-mass metamaterial concept to control the flutter onset in aircraft wings and mitigate their induced vibrations....     

基于聚合多重网格方法的跨音速颤振的数值模拟

采用流固耦合方法对跨音速颤振进行了数值模拟。流体方面在非结构网格上用有限体积方法求解了Euler方程;结构方面则求解了后掠机翼典型剖面的结构模态方程。时间推进采用双时间步长：对每一真实时间步,都通过基于聚合多重网格方法的伪时间步推进,对流体和结构方程交替迭代．得到一个稳态的流固耦合的解。文章最后给出了NACA64A010翼型剖面的跨音速颤振边界．与相关文献的计算结果符合良好。     

Aeroelastic prediction of the limit cycle oscillations of a cropped delta wing

The flutter and limit cycle oscillation (LCO) behavior of a cropped delta wing are investigated using a newly developed computational aeroelastic solver. This computational model includes a well-validated Euler finite difference solver coupled to a high-fidelity finite element structural solver. The nonlinear structural model includes geometric nonlinearities which are modelled using a co-rotational formulation. The LCOs of the cropped delta wing are computed and the results are compared to previous computations and to experiment. Over the range of dynamic pressures for which experimental results are reported, the LCO magnitudes computed using the current model are comparable to those from a previous computation which used a lower-order von Karman structural model. However, for larger dynamic pressures, the current computational model and the model which used the von Karman theory start to differ significantly, with the current model predicting larger deflections for a given dynamic pressure. This results in a LCO curve which is in better qualitative agreement with experiment. Flow features which were present in the previous computational model such as a leading edge vortex and a shock wave are enhanced in the current model due to the prediction of larger deflections and rotations at the higher dynamic pressures.     

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.     

Aeroelastic inverse: Estimation of aerodynamic loads during large amplitude limit cycle oscillations

This paper presents an algorithm to compute the aerodynamic forces and moments of an aeroelastic wing undergoing large amplitude heave and pitch limit cycle oscillations. The technique is based on inverting the equations of motion to solve for the lift and moment experienced by the wing. Bayesian inferencing is used to estimate the structural parameters of the system and generate credible intervals on the lift and moment calculations. The inversion technique is applied to study the affect of mass coupling on limit cycle oscillation amplitude. Examining the force, power, and energy of the system, the reasons for amplitude growth with wind speed can be determined. The results demonstrate that the influence of mass coupling on the pitch–heave difference is the driving factor in amplitude variation. The pitch–heave phase difference not only controls how much aerodynamic energy is transferred into the system but also how the aerodynamic energy is distributed between the degrees of freedom.     

Stability and bifurcation for limit cycle oscillations of an airfoil with external store

In this paper, stability and local bifurcation behaviors for the nonlinear aeroelastic model of an airfoil with external store are investigated using both analytical and numerical methods. Three kinds of degenerated equilibrium points of bifurcation response equations are considered. They are characterized as (1) one pair of purely imaginary eigenvalues and two pairs of conjugate complex roots with negative real parts; (2) two pairs of purely imaginary eigenvalues in nonresonant case and one pair of conjugate complex roots with negative real parts; (3) three pairs of purely imaginary eigenvalues in nonresonant case. With the aid of Maple software and normal form theory, the stability regions of the initial equilibrium point and the explicit expressions of the critical bifurcation curves are obtained, which can lead to static bifurcation and Hopf bifurcation. Under certain conditions, 2-D tori motion may occur. The complex dynamical motions are considered in this paper. Finally, the numerical solutions achieved by the fourth-order Runge–Kutta method agree with the analytic results.     

An incremental method for limit cycle oscillations of an airfoil with an external store

This paper proposes an incremental method, which is based on the harmonic balance method, to analyze the nonlinear aeroelastic problem of an airfoil with an external store. The governing equations of limit cycle oscillations (LCOs) of the airfoil are deduced by the harmonic balancing procedure. Different from usual procedures, the harmonic balance equations are not solved directly but instead transformed into an equivalent minimization problem. The minimization problem is solved using the Levenberg–Marquardt method. Numerical examples show that the LCOs obtained by the presented method are in excellent agreement with numerical solutions. The bifurcation of the LCOs is further analyzed using the Floquet theory. It is found that the LCOs exhibit saddle-node, symmetry breaking and period-doubling bifurcations with the wind speed as control parameter. Compared with the harmonic balance method, the presented method has a wider convergence region and hence makes it easier to choose a proper initial guess for iterations.     

Nonlinear state feedback control design to eliminate subcritical limit cycle oscillations in aeroelastic systems

A strictly nonlinear state feedback control law is designed for an aeroelastic system to eliminate subcritical limit cycle oscillations. Numerical continuation techniques and harmonic balance methods are employed to generate analytical estimates of limit cycle oscillation commencement velocity and its sensitivity with respect to the introduced control parameters. The obtained estimates are used in a multiobjective optimization framework to generate optimal control parameters which maximize the limit cycle oscillation commencement velocity while minimizing the control cost. Numerical simulations are used to show that the assumed nonlinear state feedback law with the optimal control parameters successfully eliminates any existing subcritical limit cycle oscillations by converting it to supercritical limit cycle oscillations, thereby guaranteeing safe operation of the system in its flight envelope.     

Construction of high-order accuracy implicit residual smoothing schemes

ＩｎｔｒｏｄｕｃｔｉｏｎＧｅｎｅｒａｌｉｍｐｌｉｃｉｔｍｅｔｈｏｄｓ (ＦｕＤ .Ｘ .[1],Ｈｉｒｓｃｈ[2 ])ｃａｎｂｅｗｒｉｔｔｅｎａｓＩｍｐｌｉｃｉｔＰａｒｔ =ＥｘｐｌｉｃｉｔＰａｒｔ . (1 )ＩｔｈａｓｂｅｅｎｐｒｏｐｏｓｅｄｂｙＭａｃＣｏｒｍａｃｋ[3]ｔｈａｔｍｏｄｅｒｎｉｍｐｌｉｃｉｔｍｅｔｈｏｄｓｃａｎｂｅｗｒｉｔｔｅｎａｓＮｕｍｅｒｉｃａｌＰａｒｔδｉＵ =ＰｈｙｓｉｃａｌＰａｒｔ . (2 )　　Ｔｈｅｐｈｙｓｉｃａｌｐａｒｔｒｅｆｌｅｃｔｓｔｈｅｃｈａｎｇｅｒｕｌｅｏｆｐｈｙｓｉｃａｌｐａｒａｍｅ…     

Numerical simulation of free surface flows with the level set method using an extremely high-order accuracy WENO advection scheme

Three-dimensional dynamic gas–liquid flow simulations that accurately track the phase interface are numerically challenging. This article presents a numerical study of the performance of the level-set phase–interface tracking method when combined with extremely high order (7th to 11th) weighted essentially non-oscillatory (WENO) advection schemes for gas–liquid free surface flows. Comparisons between simulation results and prior benchmark results suggest that such a combination of methods can be satisfactorily applied to the level-set and Navier-Stokes equations for free surface flow simulations when volume conservation is enforced at every time step, and minor numerical oscillations are suppressed through use of an artificial viscosity term. In particular, simulations of solid body rotation, the unsteady flow following an ideal dam break, tank sloshing, and the rise of a single bubble all agree with analytical or experimental results to within ± 3.12% when the level-set method is combined with an 11th order WENO scheme. Furthermore, use of an 11th order WENO advection scheme actually has a computational cost advantage because, for the same accuracy, it can be used on a coarser grid when compared with a more-common second-order advection scheme; computational savings of up to 87% are possible.     

Self-sustained oscillations and bifurcations of transonic flow past simple airfoils

A turbulent flow past two symmetric airfoils, whose bow and aft portions are circular arcs, whereas midparts are flat, is studies numerically. The amplitude of lift coefficient oscillations versus the free-stream Mach number M _∞ is analyzed at zero angle of attack. Ranges of M _∞ in which there exist flow bifurcations are determined. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 6, pp. 37–44, November–December, 2008