Mean and fluctuating components of drag and lift forces on an isolated finite-sized particle in turbulence

We perform fully resolved direct numerical simulations of an isolated particle subjected to free-stream turbulence in order to investigate the effect of turbulence on the drag and lift forces at the level of a single particle, following Bagchi and Balachandar’s work (Bagchi and Balachandar in Phys Fluids 15:3496–3513, 2003). The particle Reynolds numbers based on the mean relative particle velocity and the particle diameter are Re?=?100, 250 and 350, which covers three different regimes of wake evolution in a uniform flow: steady axisymmetric wake, steady planar symmetric wake, and unsteady planar symmetric vortex shedding. At each particle Reynolds number, the turbulent intensity is 5–10% of the mean relative particle velocity, and the corresponding diameter of the particle is comparable to or larger than the Kolmogorov scale. The simulation results show that standard drag values determined from uniform flow simulations can accurately predict the drag force if the turbulence intensity is sufficiently weak (5% or less compared to the mean relative velocity). However, it is shown that for finite-sized particles, flow non-uniformity, which is usually neglected in the case of the small particles, can play an important role in determining the forces as the relative turbulence intensity becomes large. The influence of flow non-uniformity on drag force could be qualitatively similar to the Faxen correction. In addition, finite-sized particles at sufficient Reynolds number are inherently subjected to stochastic forces arising from their self-induced vortex shedding in addition to lift force arising from the local ambient flow properties (vorticity and strain rate). The effect of rotational and strain rate of the ambient turbulence seen by the particle on the lift force is explored based on the conditional averaging using the generalized representation of the quasi-steady force proposed by Bagchi and Balachandar (J Fluid Mech 481:105–148, 2003). From the present study, it is shown that at Re?=?100, the lift force is mainly influenced by the surrounding turbulence, but at Re = 250 and 350, the lift force is affected by the wake structure as well as the surrounding turbulence. Thus, for a finite-sized particle of sufficient Reynolds number supporting self-induced vortex shedding, the lift force will not be completely correlated with the ambient flow. Therefore, it appears that in order to reliably predict the motion of a finite-sized particle in turbulence, it is important to incorporate both a deterministic component and a stochastic component in the force model. The best deterministic contribution is given by the conditional average. The influence of ambient turbulence at the scale of the particle, which are not accounted for in the deterministic contribution, can be considered in stochastic manner. In the modeling of lift force, additional stochastic contribution arising from self-induced vortex shedding must also be included.     

外部激励Stuart-Landau模型的数值研究

将非线性常微分方程组周期解的求解看作一个边值问题 ,运用Newton迭代构造求解这组方程的数值方法。利用上述方法求得了激励Stuart Landau方程的周期解 ,研究了圆柱振动对圆柱后Karman涡街的抑制现象 ,和振动的频率锁定现象 ,证明了激励Stuart Landau方程描写钝体尾迹动力系统的有效性     

Steady forces on a cylinder with oblique vortex shedding

We present a curious situation of a fluid-flow wherein the body experiences non-fluctuating fluid-flow force despite being associated with an unsteady flow comprising of sustained vortex shedding. The flow past a circular cylinder at Re=100 is investigated. It is shown that the spatio-temporal periodicity of the oblique vortex shedding results in constant-in-time force experienced by a cylinder placed in uniform flow. On the contrary, parallel vortex shedding leads to fluid force that fluctuates with time. It is found that, both, the parallel and oblique shedding are linearly unstable eigenmodes of the Re=100 steady flow past a cylinder.     

单窄条控制件对横向振荡柱体尾流 2P模式 旋涡脱落的改变

横向强迫振荡柱体尾流控制是柱体涡激振动控制的基础,在海洋、土木等工程中具有重要意义. 横向强迫振荡柱体尾流中存在一种锁频旋涡脱落模式,即在一个振荡周期内柱体上、下侧各脱落旋转方向相反的一对涡,称为2P模式. 本文将相对宽度b/D=0.32的窄条控制件置于横向强迫振荡柱体下游,对振幅比A/D=1.25, 无量纲振频f_e D/V_∞=0.22,雷诺数Re=1 200的2P模式旋涡脱落进行干扰,并通过改变控制件位置,研究旋涡的变化规律. 采用二维大涡模拟和实验验证方法进行研究,在控制件位置范围0.8≤X/D≤3.2, 0.4≤Y/D≤3.2内,得到了2P, 2S, P+S和另外6种新发现的旋涡脱落模式,并对各模式旋涡的形成过程作了详细描述. 在控制件位置平面上给出了各旋涡模式的存在区域,画出了旋涡脱落强度的等值线图,并发现在一个相当大的区域内,旋涡脱落强 度可减小一半以上,尾流变窄. 发现柱体大幅振荡引起的横向剪切流在旋涡生成中起关键作用. 探讨了控制件对横向剪切流的影响,分析了控制件在每种旋涡模式形成中的作用机制.      

圆柱绕流流场结构的大涡模拟研究

为进一步揭示绕流现象的形成机理,本文分别对处于层流稳态区、尾流过渡区、剪切层转换区Re分别为26、200、1.4×105的三种典型流态下的单圆柱绕流进行了二维数值模拟研究。Re为26时应用层流模型直接求解N-S方程,而Re分别为200、1.4×105时使用大涡模拟的方法进行计算。数值模拟很好地再现了稳定的涡旋结构、周期性交替脱落的卡门涡街结构、不规则的涡旋结构,在此基础上分析了尾流结构的基本特征及其压强分布规律、平均的流场特性、积分参数(如升力系数、阻力系数、斯特劳哈尔等),并与有关研究成果进行了对比。研究发现,采用不同流动介质时流场特性有所差异,空气为介质时的计算结果更符合实验的成果,而水为介质时计算结果偏差较大,这主要是由尾流涡旋产生的不合理负压造成的。     

The effect of air-water interface on the vortex shedding from a vertical circular cylinder

The flow past an interface piercing circular cylinder at the Reynolds number Re=2.7×10⁴ and the Froude numbers Fr=0.2 and 0.8 is investigated using large-eddy simulation. A Lagrangian dynamic subgrid-scale model and a level set based sharp interface method are used for the spatially filtered turbulence closure and the air-water interface treatment, respectively. The mean interface elevation and the rms of interface fluctuations from the simulation are in excellent agreement with the available experimental data. The organized periodic vortex shedding observed in the deep flow is attenuated and replaced by small-scale vortices at the interface. The streamwise vorticity and the outward transverse velocity generated near the edge of the separated region, which enforces the separated shear layers to deviate from each other and restrains their interaction, are primarily responsible for the devitalization of the periodic vortex shedding at the interface. The lateral gradient of the difference between the vertical and transverse Reynolds normal stresses, increasing with the Froude number, is the main source of the streamwise vorticity and the outward transverse velocity at the interface.     

Vortex-induced oscillations at low Reynolds numbers: Hysteresis and vortex-shedding modes

Results are presented for the numerical simulation of vortex-induced vibrations (VIVs) of a cylinder at low Reynolds numbers (Re). A stabilized space–time finite-element formulation is utilized to solve the incompressible flow equations in primitive variables. The cylinder, of low nondimensional mass (m^*=10), is free to vibrate in, both, the transverse and in-line directions. To investigate the effect of Re and reduced natural frequency, F_n, two sets of computations are carried out. In the first set of computations the Reynolds number is fixed (=100) and the reduced velocity (U^*=1/F_n) is varied. Hysteresis, in the response of the cylinder, is observed at the low- as well as high-end of the range of reduced velocity for synchronization/lock-in. In the second set of computations, the effect of Reynolds number (50Re500) is investigated for a fixed reduced velocity (U^*=4.92). The effect of the Reynolds number is found to be very significant for VIVs. While the vortex-shedding mode at low Re is 2S (two single vortices shed per cycle), at Re300 and larger, the P+S mode of vortex shedding (a single vortex and one pair of counter-rotating vortices are released in each cycle of shedding) is observed. This is the first time that the P+S mode has been observed for a cylinder undergoing free vibrations. This change of vortex-shedding mode is hysteretic in nature and results in a very large increase in the amplitude of in-line oscillations. Since the flow ceases to remain two-dimensional beyond Re200, it remains to be seen whether the P+S mode of shedding can actually be observed in reality for free vibrations.     

Suppression of lift fluctuations on a circular cylinder by inducing the symmetric vortex shedding mode

The flow around a stationary circular cylinder modified by two synthetic jets positioned at the mean separation points is numerically studied. The Reynolds number based on the free-stream velocity and the circular cylinder diameter is Re=500. The focus is to present a novel way to suppress the lift fluctuations by changing the vortex shedding mode, and thus particular attention is paid to the interactions between the synthetic jets and wake shear layers and the resulting vortex dynamics. The overall influences of both momentum coefficient and excitation frequency are discussed. In some simulated cases, the vortex lock-on phenomenon is discovered, which causes the typical Kàrmàn type vortex shedding to be converted into the symmetric shedding modes, leading to the complete suppression of lift fluctuations. In other cases, the asymmetric shedding mode still dominates the wake evolution. Detailed vortical evolution for each typical wake pattern is analyzed to reveal the control mechanism. Additionally, the control effectiveness is evaluated, indicating that the present control strategy contributes an effective way to suppress the lift fluctuations and reduce the mean drag.     

旋转振动圆柱绕流周期解和Floquet稳定性

对低雷诺数旋转振动圆柱绕流问题运用低维Galerkin方法将N－S方程约化为一组非线性常微分方程组。运用打靶法数值求解了这组方程的周期解,并用Tloquet理论对周期解的稳定性进行了分析,确定了流动失稳的机制。     

Two- and three-dimensional numerical simulations of the clap–fling–sweep of hovering insects

The importance of three-dimensional effects for flapping wings is addressed by means of numerical simulation. In particular, the clap–fling–sweep mechanism is examined. The flow at the beginning of the downstroke is shown to be in reasonable agreement with the two-dimensional approximation. After the wings move farther than one chord length apart, three-dimensional effects become essential. Two values of the Reynolds number are considered. At Re=128, the spanwise flow from the wing roots to the wing tips is driven by the centrifugal forces acting on the mass of the fluid trapped in the recirculation bubble behind the wings. It removes the excess of vorticity and delays the periodic vortex shedding. At Re=1400, vortex breakdown occurs past the outer portion of the wings, and multiple vortex filaments are shed into the wake.     

Numerical simulation of flow around a square cylinder in uniform-shear flow

This paper presents results obtained from a numerical simulation of a two-dimensional (2-D) incompressible linear shear flow over a square cylinder. Numerical simulations are performed, using the lattice Boltzmann method, in the ranges of 50⩽Re⩽200 and 0⩽K⩽0.5, where Re and K are the Reynolds number and the shear rate, respectively. The effect of the shear rate on the frequency of vortex shedding from the cylinder, and the lift and drag forces exerted on the cylinder are quantified together with the flow patterns around the cylinder. The present results show that vortex structure behind the cylinder is strongly dependant on both the shear rate and Reynolds number. When Re=50, a small K can disturb the steady state and cause an alternative vortex shedding with uneven intensity. In contrast, a large value of K will suppress the vortex shedding from the cylinder. When Re>50, the differences in the strength and size of vortices shed from the upper and lower sides of the cylinder become more pronounced as K increases. Vortex shedding disappears when K is larger than a critical value, which depends on Re. The flow patterns around the cylinder for different Re tend towards self-similarity with increasing K. The lift and drag forces exerted on the cylinder, in general, decrease with increasing K. Unlike a shear flow past a circular cylinder, the vortex shedding frequency past a square cylinder decreases with increasing the shear rate. A significant reduction of the drag force occurs in the range 0.15<K<0.3.     

Effects of aspect ratio and orientation on the wake characteristics of low Reynolds number flow over a triangular prism

The wake characteristics of unconfined flows over triangular prisms of different aspect ratios have been numerically analysed in the present work. For this purpose, a fixed Cartesian-grid based numerical technique that involves the porous medium approach to mimic the effect of solid blockage has been utilised. Correspondingly, laminar flow simulations ranging from the sub-critical regime (before the onset of vortex shedding) to the super-critical regime have been considered here within the limits of two-dimensionality. In the sub-critical regime, correlations relating the wake bubble length with Reynolds number (Re) have been proposed for various aspect ratios. Also, the effects of aspect ratio and Reynolds Number on the drag force coefficient (C_D) have been characterised for two different geometrical orientations of the prism (base or apex facing the flow). Subsequently, the critical Reynolds number at the onset of vortex shedding has been predicted for each of the aspect ratio considered, by an extrapolation procedure. The unsteady flow characteristics of the super-critical regime are finally highlighted for different aspect ratios and triangular orientations considered in the study.     

Vibrations of a cylinder in a uniform flow in the presence of a no-slip side-wall

A circular cylinder placed in a uniform flow, and that spans the entire length between two side walls, may experience either parallel or oblique vortex shedding depending on the end conditions. It was shown by Mittal and Sidharth (2014) that the spatio-temporal periodicity of the oblique vortex shedding results in constant-in-time force experienced by the cylinder. On the contrary, parallel vortex shedding leads to fluid force that fluctuates with time. The free vibrations of a circular cylinder, in the presence of a wall, are investigated. For comparison, computations with end walls, where a slip condition on velocity is specified, are also carried out. The Reynolds number, based on the diameter of the cylinder and free-stream speed of the flow, is Re=100. The initial condition for the free vibrations is the fully developed unsteady flow past a stationary cylinder with oblique shedding. It is found that as the amplitude of vibration of the cylinder builds up, the vortices shed from the cylinder align with its axis leading to parallel shedding. The response of the cylinder is associated with two branches: initial and lower. On the lower branch, the response of the cylinder is virtually identical from two- and three-dimensional computations. The flow as well as the response is different on the initial branch and outside the synchronization regime. Forced vibrations confirm the phenomena.     

Reduced‐order controllers for control of flow past an airfoil

Reduced‐order controller design by means of reduced‐order model for control of a wake flow is presented. Reduced‐order model is derived by combining the Galerkin projection with proper orthogonal decomposition (POD) or with other related reduced‐order approaches such as singular value decomposition or reduced‐basis method. In the present investigation, we discuss the applicability of the reduced‐order approaches for fast computation of the optimal control for control of vortex shedding behind a thin airfoil through unsteady blowing on the airfoil surface. Accuracy of the reduced‐order model is quantified by comparing flow fields obtained from the reduced‐order models with those from the full‐order simulations under the same free‐stream conditions. A control of vortex shedding is demonstrated for Reynolds number 100. It is found that downstream directed blowing on the upper surface of the airfoil near the leading edge is more efficient in mitigating flow separation and suppressing the vortex shedding. Copyright © 2005 John Wiley & Sons, Ltd.     

Experimental study on the confined 3D laminar flow past a square prism with a high blockage ratio

An experimental PIV study is presented that addresses the confined 3D laminar flow behaviour past a square prism. The Reynolds number (Re), based on prism cross-section height varies between 100 and 256. The channel aspect ratio is 1/1 and the blockage ratio is 1/2.5. This geometry is representative of a passive method to enhance mixing in otherwise highly ordered laminar channel flow. It is found that the lateral walls exert a strong effect on the flow behaviour with two main consequences: (a) the onset of vortex shedding is delayed to a Re in the vicinity of 170, as opposed to the unconfined case where the critical Re is reported to be between 50 and 60 and (b) transition from the steady closed recirculation bubble regime to the vortex shedding regime is not abrupt. In particular, there is a range of Re for which the closed recirculation bubble pulsates with increasing amplitude prior to the onset of the Karman street regime. The experimental results are supported by numerical computations, and the relation of the results with the practical design of engineering systems is also discussed.     

低雷诺数下串列三圆柱涡激振动中的弛振现象及其影响因素

对间距比为1.2和雷诺数为100的串列三圆柱涡激振动进行数值模拟, 发现在某个折合流速之后, 三圆柱的响应均呈现为随着折合流速增大而增大的弛振现象, 平衡位置偏移、低频振动以及旋涡脱落与圆柱运动之间的时机三个因素共同决定了弛振现象的出现. 进一步的研究发现, 串列三圆柱的弛振现象仅出现在质量比不大于2.0和雷诺数不大于100的工况下. 当质量比较大时, 串列三圆柱的平衡位置固定不变, 且圆柱的振动不规律, 使得旋涡脱落与圆柱运动的时机处于变化之中. 当雷诺数较高时, 最上游圆柱的平衡位置在折合流速较大时回到初始位置, 不再参与对圆柱振动的调节, 使得圆柱的振动响应不再规律, 旋涡脱落与圆柱运动的时机也一直处于变化之中.      

横向振荡圆柱绕流的格子Boltzmann方法模拟

基于格子Boltzmann方法(LBM)对不可压横向振荡圆柱绕流问题进行了数值研究. 与传统的求解宏观的N-S方程的数值方法不同, LBM求解此类问题不需要采用动网格, 而且不需要对网格进行特殊处理, 从而节约了计算成本. 结果显示, 当振荡频率增加到相应的静止圆柱绕流的自然涡脱落频率附近时, 圆柱后最新形成的集中涡距离柱体越来越近, 直到达到一个极限位置. 随后, 集中涡突然转向圆柱体另一侧脱落. 当振荡频率接近于静止圆柱的自然涡脱落频率时, 发生频率同步的现象. 随着振荡频率远离自然涡脱落频率, 同步现象消失. 在几种次谐振荡和超谐振荡下, 尾流区的涡脱落频率仍为相应的静止圆柱绕流的自然涡脱落频率.      

Effect of novel synthetic jet on wake vortex shedding modes of a circular cylinder

A novel actuator signal achieved by changing the ratio of the suction duty cycle to the blowing duty cycle is adopted to enhance the control effect of the synthetic jet for the flow around a circular cylinder. The suction duty cycle factor k defined as the ratio between the time duration of the suction cycle and the blowing cycle and the equivalent momentum coefficient C_μ are introduced as the determining parameters. The synthetic jet is positioned at the rear stagnation point in order to introduce symmetric perturbations upon the flow field. The proper orthogonal decomposition (POD) technique is applied for the analysis of the spanwise vorticity field. Increasing the suction duty cycle factor, the momentum coefficient is enhanced, and thus a stronger and larger scale synthetic jet vortex pair with a higher convection velocity is generated. The synthetic jet vortex pair interacts with the spanwise vorticity shear layers behind both sides of the cylinder, resulting in the variations of the wake vortex shedding modes at Re=950: for k=0.25, C_μ=0.148, vortex synchronization at the subharmonic excitation frequency with antisymmetric shedding mode; for 0.50≤k≤1.00, 0.213≤C_μ≤0.378, vortex synchronization at the excitation frequency with the symmetric or antisymmetric shedding modes; for 2.00≤k≤4.00, 0.850≤C_μ≤2.362, vortex synchronization at the excitation frequency with symmetric shedding mode. Hence, the control effect of the synthetic jet upon the wake vortex of a circular cylinder can be enhanced by increasing the suction duty cycle factor so as to increase the momentum coefficient. This is also validated at a higher Reynolds number Re=1600.     

Analyzing the fluctuating pressures acting on a circular cylinder using stochastic decomposition

Fluctuating wind pressures acting on bluff bodies are influenced by approaching turbulence and signature (body-induced) turbulence. For a circular cylinder, the signature turbulence is closely related to the formation of Karman vortex shedding. In this paper, proper orthogonal decomposition (POD) and spectral proper transformation techniques (SPT) are applied to the pressure fluctuations acting on a circular cylinder. The physical relationships between the decomposed modes and vortex shedding are discussed to identify the dominant aerodynamic behavior (lift or drag) and to evaluate its contribution to overall behavior. The effect of Reynolds number (Re) is also addressed. It is found that the application of POD and SPT can separate the along-wind and across-wind effects on the cylinder model in both subcritical and supercritical regimes. In contrast to POD, the SPT mode is formulated in the frequency domain, and the dynamic coherent structures can be defined in terms of amplitude and phase angle, which allows detection of the advection features of vortex shedding. In addition, it is observed that the energy contribution of the shedding induced lift force increases with Re and gradually becomes a dominant aerodynamic force at Reynolds numbers in the supercritical regime.