Flutter of slender bodies under axial stress

The equations of motion of flexible slender bodies with constant body sections immersed in a uniform axial flow are discussed and used to derive some simple results for the divergence speed and the flutter speed. The results are compared with a classical waving flag result in two-dimensional flow. The slender body result for the flutter speed is compared with values obtained from wind tunnel experiments for some low budget paper strips.     

Parametric studies on relationships between flutter derivatives of slender bridge (Ⅱ)

Based on curve fitting of coefficients of three component forces of the Messina Straits Bridge, and the previously proposed semi-analytical expressions of flutter deriva-tives of flexible structure, the change of flutter derivatives of slender bridge cross-section with respect to its aerodynamic center, rotational speed and angle variation is studied using a parametric method. The calculated results are compared with the measured ones, and expressions of flutter derivatives of the Messina Straits Bridge are derived. The in-trinsic relationships existing in flutter derivatives are validated again. It is shown that the influence of the rotational speed on flutter derivatives is not negligible. Therefore, it provides an additional semi-analytical approach for analyzing flutter derivatives of the bridge with streamlined cross-section to get its aerodynamic information.     

Parametric studies on relationships between flutter derivatives of slender bridge （Ⅱ）

Based on curve fitting of coefficients of three component forces of the Messina Straits Bridge, and the previously proposed semi-analytical expressions of flutter derivatives of flexible structure, the change of flutter derivatives of slender bridge cross-section with respect to its aerodynamic center, rotational speed and angle variation is studied using a parametric method. The calculated results are compared with the measured ones, and expressions of flutter derivatives of the Messina Straits Bridge are derived. The intrinsic relationships existing in flutter derivatives are validated again. It is shown that the influence of the rotational speed on flutter derivatives is not negligible. Therefore, it provides an additional semi-analytical approach for analyzing flutter derivatives of the bridge with streamlined cross-section to get its aerodynamic information.     

Parametric studies on relationships between flutter derivatives of slender bridge （Ⅰ）

Relationships between flutter derivatives of slender bridge are investigated based on our previously proposed semi-analytical flutter derivatives of flexible structure. The intrinsic relations are validated with test data of flutter derivatives of two bridges. Changes in flutter derivatives with the aerodynamic center, rotation speed, and angle variation are also studied by using a parametric method. The results show correctness of the proposed expressions of flutter derivatives given by authors in Ref. [1], and indicate that certain relations exist between these derivatives. It is also shown that semi-analytical flutter derivatives are applicable to bridges with a streamlined cross-section.     

水下亚声速细长锥型射弹超空泡流的数值计算方法

采用理想可压缩流体无旋定常流动及超空泡尾部Riabushinsky闭合方式假定,基于水动力学势流理论及细长体理论,建立了描述水下亚声速条件下细长锥型射弹超空泡流动的积分微分方程。发展了求解该方程的数值离散方法,提出多种超空泡外形初始解,分析了它们对超空泡形态计算结果的影响,优化了计算过程,简化了初始迭代条件。分析了流体压缩性对超空泡流动参数的影响,当马赫数大于0.3时,超空泡外形、射弹表面压力系数及射弹运动压差阻力系数均明显增大。计算得到的超空泡流动参数与相关文献的理论和实验结果吻合良好。     

细长体水下运动空化流场及弹道特性实验

为了研究细长体水下高速运动时空泡的产生、闭合及脱落特性,及影响细长体空泡形态及弹道特性的复杂因素等,初步开展了细长体模型水下高速运动的实验研究,分析了不同初始空化数下细长体模型在水中高速运动的一系列流动现象,重点研究了空泡的发展、闭合、尾部回射流和尾部脱落特性,以及轴对称细长体模型弹道特性与空泡形态变化之间的关系和转动特性随时间的变化历程等。结果表明:细长体水下高速运动时形成超空泡,空泡头部光滑透明,尾部凝结有汽水混合物且有交替脱落的含气漩涡;初始空化数对细长体的速度衰减有所影响;受初始扰动影响,细长体水下运动伴随有绕头部的转动且初始扰动影响细长体俯仰角随时间的变化历程。     

An improved theoretical model for the dynamics of a free–clamped cylinder in axial flow

A new, improved linear analytical model is presented in this paper for the dynamics of a slender cantilevered cylinder with an ogival free end, subjected to axial flow directed from its free end towards the clamped one. In the present model the fluid-dynamic forces at the free end of the cylinder are analysed in a meticulous manner. The model predicts that the cylinder loses stability at relatively low flow velocity by flutter, and then at higher flow velocity by static divergence. This agrees with the dynamical behaviour observed in experiments. Moreover, quantitative agreement in the critical flow velocities for flutter and divergence between this improved theory and experiment is fairly good.     

细长体大迎角非对称涡流的数值研究

通过数值方法对大迎角细长体低速湍流流场的模拟,探讨头部顶端极小扰动对细长体非对称绕流形成与发展的影响.结果表明在细长体顶端附近施加极小扰动可以模拟出实验观测到的非对称流场,非对称的涡系结构沿轴向是逐步发展的,截面侧向力沿轴向的分布呈现正弦型曲线的变化特征,扰动能量经过指数增长后达到饱和,有效扰动的规模影响涡流非对称性的大小及分布,单侧扰动产生的流场非对称性随扰动周向位置的变化呈现单周期性规律.小扰动诱发非对称的数值算例表明非对称绕流的形成是源于流场的空间不稳定性机制.     

CHAOTIC MOTIONS AND LIMIT CYCLE FLUTTER OF TWO-DIMENSIONAL WING IN SUPERSONIC FLOW

Based on the piston theory of supersonic flow and the energy method, the flutter motion equations of a two-dimensional wing with cubic stiffness in the pitching direction are established. The aeroelastic system contains both structural and aerodynamic nonlinearities. Hopf bifurcation theory is used to analyze the flutter speed of the system. The effects of system parameters on the flutter speed are studied. The 4th order Runge-Kutta method is used to calculate the stable limit cycle responses and chaotic motions of the aeroelastic system. Results show that the number and the stability of equilibrium points of the system vary with the increase of flow speed. Besides the simple limit cycle response of period 1, there are also period-doubling responses and chaotic motions in the flutter system. The route leading to chaos in the aeroelastic model used here is the period-doubling bifurcation. The chaotic motions in the system occur only when the flow speed is higher than the linear divergent speed and the initial condition is very small. Moreover, the flow speed regions in which the system behaves chaos axe very narrow.     

Experimental investigation of side force control on cone-cylinder slender bodies with flexible micro balloon actuators

Side forces on slender bodies of revolution at medium to high angles of attack (AOA > 30°) has been known from a large number of investigations. Asymmetric vortex pairs over a slender body are believed to be the principle cause of the side forces. Under some flight conditions, this side force may be as large as the normal force acting on the slender body. In this paper, experimental results are presented for side force control on a cone-cylinder slender body by using microfabricated balloon actuators. The micro balloon actuators are made of polydimethylsiloxane (PDMS) elastomer by using micro molding techniques. They can be packaged on curve surfaces of a cone-cylinder slender body. As actuator is actuated, the micro balloon actuator inflates about 1.2 mm vertically, which is about 2.4% of the cylinder diameter D (=50 mm) of the cone-cylinder slender body. Micro balloon actuators are actuated at different roll angles of a cone-cylinder slender body. Aerodynamic force measurement results indicate the effects of micro balloon actuators vary at different actuation locations on the cone-cylinder slender body. The side forces can be significantly reduced if the actuators are actuated in the weak vortex side (the side corresponding to the asymmetric vortex which is far from the surface) and actuation angles are located at about 50–60° (the actuation angle here is measured from stagnation line of the incidence plane toward weak vortex side direction). Significant changes are noticed from the surface pressure, as well as leeside vortex flow field, measurement. Micro balloon actuators change nose shapes of the slender body which decide adverse-pressure-gradient values and directly influence the origin of the separation lines and characteristics of the separated vortices over the leeside surface.     

A flexible rod in annular leakage flow: Influence of turbulence and equilibrium offset,and analysis of instability mechanisms

The linear stability of a flexible, cylindrical rod subjected to annular leakage flow is studied. The mathematical models developed by Li, Kaneko, and Hayama in 2002 and Fujita and Shintani in 2001 are bridged and extended, to account for a flexible rod with equilibrium offset (eccentricity) in laminar or turbulent leakage flow. Stability characteristics are analyzed numerically for a variety of configurations. It is found that simply supported rods may become unstable at a certain critical flow speed by either divergence or flutter, depending on dimensions and fluid/solid properties. It is furthermore found that the critical flow speed is quite insensitive to use of a laminar friction model at high Reynolds numbers in cases of divergence, but sensitive to it in cases of flutter. These findings are verified analytically though analysis of an energy equation. This equation shows that (i) divergence instability is independent of fluid friction; (ii) flutter instability is caused solely by fluid friction. It also suggests a possible explanation to the question of why a ‘wrong’ fluid friction assumption gives a too large critical flow speed in cases of flutter instability at a high Reynolds number.     

Influence of nose perturbations on behaviors of asymmetric vortices over slender body

The influence of nose perturbations on the behaviors of asymmetric vortices over a slender body with a three-caliber ogive nose is studied in this paper. The tests of a nose-disturbed slender body with surface pressure measurement were conducted at a low speed wind tunnel with subcritical Reynolds number of 1×10⁵ at angle of attack α=50°. The experiment results show that the behaviors and structure of asymmetric vortices over the slender body are mainly controlled by manual perturbation on the nose of body as compared with geometrical minute irregularities on the test model from the machining tolerances. The effect of the perturbation axial location on asymmetric vortices is the strongest if its location is near the model apex. There are four sensitive circumferential locations of manual perturbation at which bistable vortices over the slender body are switched by the perturbation. The flowfield near the reattachment line of lee side is more sensitive to the perturbation, because the saddle point to saddle point topological structure in this reattachment flowfield is unstable. Various types of perturbation do not change the perturbation effect on the behaviors of bistable asymmetric vortices. The project supported by the National Natural Science Foundation of China (10172017) and the Foundation of National Key Laboratory of Aerodynamic Design and Research (00JS51.3.2 HK01)     

Optimization of the shape of a body with respect to two criteria: Radiation heat flux and wave drag

The numerical solution of the two-criteria variational problem of the body contour with minimum radiation heat flux and wave drag is obtained in the class of axisymmetric and plane slender bodies in hypersonic flow. Solutions obtained using the Pareto, ideal point and minimax methods are compared. It is shown that in the class of axisymmetric slender bodies the optimum body gives a decrease in the radiation heat flux as compared with a cone of up to 15% for the Pareto method, up to 13% for the ideal point method, and up to 5% for the minimax method. A solution is also obtained in the subclass of power-law slender bodies and it is shown that the optimum power-law bodies are inferior, as compared with the optimum bodies from the general class of such bodies, in reducing both radiation heating and resistance.     

细长体后部非定常超空泡研究

采用积分方程方法,研究了轴对称细长体后部非定常超空泡问题.应用时间有限差分离散化方法,对积分方程进行了求解.以细长锥体空化器为例,文中分别给出了当锥角和空化数改变(简称扰动)时,空泡长度和形状的变化规律.当流场周期扰动时,分析计算了超空泡的尺度变化.分别采用本文方法和理论公式,对空泡长度与空化数的关系曲线进行了对比.数值结果表明,扰动周期越短,空泡长度的变化越小;在相同的扰动频率下,空泡越长,时间滞后越长;空泡长度相同时,扰动频率越高,时间滞后越长.在高频脉冲扰动下,有脉冲波形沿着空泡表面传播,其传播速度为来流速度.在周期小扰动情况下,扰动波形沿着空泡表面传播,传播速度也是来流速度.本文得到的数值结果为水下航行体空化器的分析和设计提供参考作用.     

Analysis of self-excited forces for a box-girder bridge deck through unsteady RANS simulations

This paper presents the results of two-dimensional numerical simulations of the flow field around a trapezoidal box-girder bridge section with later cantilevers, experiencing small-amplitude heaving or pitching harmonic oscillations. Unsteady Reynolds-averaged Navier–Stokes equations are solved in conjunction with an eddy-viscosity and an explicit algebraic Reynolds stress model. Flutter derivatives are determined and compared with wind tunnel results, showing fairly good agreement. The degree of sharpness of the deck lower edges is found to play a key role in the aeroelastic behavior of the profile. In particular, the bridge section fully behaves as a bluff body and is prone to low-reduced-wind-speed torsional galloping in the case of perfectly sharp edges. By contrast, the presence of a small radius of curvature in the section lower corners changes the nature of the instability to coupled flutter and significantly postpones the stability threshold, in line with a quasi-streamlined body behavior. Moreover, a wide sensitivity study is carried out, investigating the influence on the self-excited forces of the amplitude of oscillation, mean angle of attack and Reynolds number. In particular, the numerical simulations for the geometry with smooth lower edges highlight the regime transition occurring when the Reynolds number is varied, with significant effects on the flutter derivatives. Finally, the numerical flow visualizations provide a physical explanation of some phenomena observed in the wind tunnel experiments.     

高速小展弦比机翼颤振的微分求积法分析

引入微分求积法,分析高速小展弦比机翼的气动弹性问题。将小展弦比机翼等效为悬臂板,基于一阶活塞气动力理论建立机翼颤振偏微分方程,采用微分求积法将偏微分方程转化为常微分方程,根据频率重合理论对颤振问题进行求解。分析了机翼的固有频率及颤振速度,并与有限元软件计算结果进行比较,误差在2％以内,很好的验证了微分求积法求解小展弦比机翼颤振问题的有效性。分析了机翼面积、展弦比及厚度对颤振速度的影响,结果表明,小展弦比机翼的颤振速度受结构尺寸的影响较大,颤振速度随面积和展弦比的增大而减小,随机翼厚度的增大而增大。     

Flutter of a viscoelastic plate in a supersonic gas flow

The flutter of a viscoelastic plate in a supersonic gas flow is studied. A technique and algorithm for numerical solution of nonlinear integro-differential equations with weakly singular kernels are elaborated. The critical flutter speed of viscoelastic plates is determined     

Motion of a slender body in quiescent polymer solutions

The motion of a slender body falling in quiescent polymer solutions is investigated experimentally. It represents the simplest model of motion of single fibers in the flow of fiber suspensions. The fall behavior in quiescent polymer solutions is compared with that in water. It is demonstrated that a slender body falling in Newtonian liquids rotates to adopt a horizontal orientation, whereas in non-Newtonian liquids it rotates towards a vertical orientation but for less concentrated solutions is not able to reach the vertical orientation and moves sideways with a constant orientation angle. The effects of shear thinning and elasticity on the motion of the body are discussed.