Numerical computation of a three-dimensional vortex sheet in a swirl flow

We consider an incompressible and inviscid fluid flow, called “swirl flow” that rotates around a certain axis in three-dimensional space. We investigate numerically the dynamics of a three-dimensional vortex sheet which is defined as a surface across which the velocity field of the swirl flow changes discontinuously. The vortex method and a fast summation method are implemented on a parallel computer. These numerical methods make it possible to compute the evolution of the vortex sheet for a long time and to describe the complex dynamics of the sheet.     

On Low-Dimensional Galerkin Models for Fluid Flow

In this paper some implications of the technique of projecting the Navier–Stokes equations onto low-dimensional bases of eigenfunctions are explored. Such low-dimensional bases are typically obtained by truncating a particularly well-suited complete set of eigenfunctions at very low orders, arguing that a small number of such eigenmodes already captures a large part of the dynamics of the system. In addition, in the treatment of inhomogeneous spatial directions of a flow, eigenfunctions that do not satisfy the boundary conditions are often used, and in the Galerkin projection the corresponding boundary conditions are ignored. We show how the restriction to a low-dimensional basis as well as improper treatment of boundary conditions can affect the range of validity of these models. As particular examples of eigenfunction bases, systems of Karhunen–Loève eigenfunctions are discussed in more detail, although the results presented are valid for any basis. Received 10 September 1999 and accepted 13 December 1999     

Vortex dynamics of a sphere wake in proximity to a wall

Toward getting the vortex dynamics characteristics and wake structure of a sphere in proximity to a wall, the effect of a proximal flat plate on the wake of a stationary sphere is investigated via direct numerical simulation. The vortex shedding process and the significant variation of the wake structure are described in detail. The drag coefficient reduces and the wake structure of the sphere becomes complex due to the combined effect of the wake flow and the wall. A jet flow forms between the sphere and the flat plate, which suppresses the vortex separation on the bottom of the sphere. The asymmetric distributions of the coherent structures and the recirculation region behind the sphere are discussed. Besides vortex shedding patterns, the time-averaged velocity distribution, vortex dynamics, distribution regularities of turbulent kinetic energy and enstrophy are investigated.     

Numerical study of vortex interactions behind two circular cylinders in bistable flow regime

A two-dimensional discrete-vortex model was used to investigate vortex interaction inside the near wakes of two circular cylinders in side-by-side arrangement within bistable flow regime. Two phases of vortex evolution are mainly identified in the near wakes: a symmetric shedding phase, characterized by two antiphase vortex streets, and a flip-flopping phase, characterized by biased gap flow, switching at irregular intervals. For the flip-flopping phase, vortex amalgamation, vortex pairing and dipole are found. Vortex dynamics of the flow is presented and its possible effects on the flow parameters are discussed. The initiation and transition from the symmetric to flip-flopping phase are caused by the asymmetry of one of the gap vortices. Flow visualization and quantitative results obtained seem to support the findings from the model.     

Spatiotemporal complexity of the aortic sinus vortex

The aortic sinus vortex is a classical flow structure of significant importance to aortic valve dynamics and the initiation and progression of calcific aortic valve disease. We characterize the spatiotemporal characteristics of aortic sinus vortex dynamics in relation to the viscosity of blood analog solution as well as heart rate. High-resolution time-resolved (2 kHz) particle image velocimetry was conducted to capture 2D particle streak videos and 2D instantaneous velocity and streamlines along the sinus midplane using a physiological but rigid aorta model fitted with a porcine bioprosthetic heart valve. Blood analog fluids used include a water–glycerin mixture and saline to elucidate the sensitivity of vortex dynamics to viscosity. Experiments were conducted to record 10 heart beats for each combination of blood analog and heart rate condition. Results show that the topological characteristics of the velocity field vary in timescales as revealed using time bin-averaged vectors and corresponding instantaneous streamlines. There exist small timescale vortices and a large timescale main vortex. A key flow structure observed is the counter vortex at the upstream end of the sinus adjacent to the base (lower half) of the leaflet. The spatiotemporal complexity of vortex dynamics is shown to be profoundly influenced by strong leaflet flutter during systole with a peak frequency of 200 Hz and peak amplitude of 4 mm observed in the saline case. While fluid viscosity influences the length and timescales as well as the introduction of leaflet flutter, heart rate influences the formation of counter vortex at the upstream end of the sinus. Higher heart rates are shown to reduce the strength of the counter vortex that can greatly influence the directionality and strength of shear stresses along the base of the leaflet. This study demonstrates the impact of heart rate and blood analog viscosity on aortic sinus hemodynamics.     

Near field vortex dynamics in axially forced,laminar, co-flowing jets: a descriptive study of the flow configurations

An experimental study is presented of the vortex structures that appear in the shear layer of laminar, co-flowing air jets subjected to strong axial forcing. A set of flow visualisation experiments has been performed to elucidate the nature of the different structures and the mechanisms leading to their appearance and further interactions. The axial forcing sets the axisymmetric instability to prescribed values of amplitude and frequency (and thus wavelength) and produces a strong effect in the lateral spreading of the inner jet. It is shown that the near field development of the flow can be explained via inviscid vortex dynamics arguments, involving three vortex structures. Due to the strong axial forcing, all these vortices already appear as developed concentrations of vorticity in the surroundings of the nozzle exit. An azimuthal perturbation is added to the flow in the form of a lobed nozzle exit, in order to lock the azimuthal organisation of the vortices. The results are discussed and some representative configurations are examined. Each configuration appears for a given range of the forcing parameters. A tentative model of the near-field vortex dynamics is developed, but quantitative measurements are still necessary.     

Numerical simulation of flapping‐wing insect hovering flight at unsteady flow

A computational fluid dynamics (CFD) analysis was conducted to study the unsteady aerodynamics of a virtual flying bumblebee during hovering flight. The integrated geometry of bumblebee was established to define the shape of a three‐dimensional virtual bumblebee model with beating its wings, accurately mimicking the three‐dimensional movements of wings during hovering flight. The kinematics data of wings documented from the measurement to the bumblebee in normal hovering flight aided by the high‐speed video. The Navier–Stokes equations are solved numerically. The solution provides the flow and pressure fields, from which the aerodynamic forces and vorticity wake structure are obtained. Insights into the unsteady aerodynamic force generation process are gained from the force and flow‐structure information. The CFD analysis has established an overall understanding of the viscous and unsteady flow around the virtual flying bumblebee and of the time course of instantaneous force production, which reveals that hovering flight is dominated by the unsteady aerodynamics of both the instantaneous dynamics and also the past history of the wing. A coherent leading‐edge vortex with axial flow and the attached wingtip vortex and trailing edge vortex were detected. The leading edge vortex, wing tip vortex and trailing edge vortex, which caused by the pressure difference between the upper and the lower surface of wings. The axial flow, which include the spanwise flow and chordwise flow, is derived from the spanwise pressure gradient and chordwise pressure gradient, will stabilize the vortex and gives it a characteristic spiral conical shape. Copyright © 2006 John Wiley & Sons, Ltd.     

Uncovering large-scale coherent structures in natural and forced turbulent wakes by combining PIV,POD, and FTLE

Planar velocity data of the unsteady separated flow in the turbulent wake of a circular cylinder obtained by particle image velocimetry (PIV) are analyzed in order to visualize the large-scale coherent structures associated with alternating vortex shedding at a Reynolds number of 2,150. Two different cases are examined: unforced vortex shedding in the natural wake and vortex lock-on incited by forced perturbations superimposed in the inflow velocity. Proper orthogonal decomposition (POD) is employed to reconstruct the low-order wake dynamics from randomly sampled snapshots of the velocity field. The reconstructed flow is subsequently used to determine the evolution of the finite-time Lyapunov exponent (FTLE) fields which identify the Lagrangian coherent structures. The results demonstrate that the combination of methods employed offers a powerful visualization tool to uncover large-scale coherent structures and to exemplify vortex dynamics in natural and forced bluff-body wakes.     

Coherent and turbulent processes in the bistable regime around a tandem of cylinders including reattached flow dynamics by means of high-speed PIV

The turbulent flow around two cylinders in tandem at the sub-critical Reynolds number range of order of 10⁵ and pitch to diameter ratio of 3.7 is investigated by using time-resolved Particle Image Velocimetry (TRPIV) of 1 kHz and 8 kHz. The bi-stable flow regimes including a flow pattern I with a strong vortex shedding past the upstream and the downstream cylinder, as well as a flow pattern II corresponding to a weak alternating vortex shedding with reattachment past the upstream cylinder are investigated. The structure of this “reattachment regime” has been analyzed in association with the vortex dynamics past the downstream cylinder, by means of POD and phase-average decomposition. These elements allowed interconnection among all the measured PIV planes and hence analysis of the reattachment structure and the flow dynamics past both cylinders. The results highlight fundamental differences of the flow structure and dynamics around each cylinder and provide the ‘gap’ flow nature between the cylinders. Thanks to a high-speed camera of 8 kHz, the shear-layer vortices tracking has been possible downstream of the separation point and the quantification of their shedding frequency at the present high Reynolds number range has been achieved. This issue is important regarding fluid instabilities involved in the fluid–structure interaction of cylinder arrays in nuclear reactor systems, as well as acoustic noise generated from the tandem cylinders of a landing gear in aeronautics.     

Turbulent flow dynamics caused by a truncated cylinder

The turbulent flow field around a quite simple geometry has been analysed in detail based on a snapshot database taken from numerical simulation. Here, emphasis is placed on the dominant coherent motion and the flow dynamics in the separated wake. The method-based analysis is performed using POD, filtering and phase-averaging. The results obtained show a highly intermittent flow topology, which reveals different (at least three) recurring vortex arrangements, but with considerably stochastic character. Corresponding frequencies, the periodicity as well as correlation and interaction of predominant vortex motions are discussed. The methods employed are not limited to the configuration exemplarily chosen.     

Formation, evolution, and decay of a vortex street in the wake of a streamlined body

The decay of a Kármán vortex street and the formation of a secondary vortex structure in the far wake of a streamlined cylinder are studied. The dynamics of spatially evolving vortex structures is examined in the free flow and in the following ways of external influence on this flow: rotation with a constant velocity and translational and rotational oscillations of the cylinder. The results are obtained by numerically solving the Navier-Stokes equations with two different methods. The corresponding boundary value problems are formulated in the domains extended up to 500 radii of the cylinder.     

Vortices in time-periodic shear flow

Vortices emerging in geophysical turbulence may experience deformations due to the non-uniform ambient flow induced by neighbouring vortices. At first approximation this ambient flow is modeled by a linear shear flow. It is well known from previous studies that the vortex may be (partially) destructed through removal of weak vorticity at the vortex edge—a process referred to as ‘stripping’. While most previous studies considered a stationary external shear flow, we have examined the behaviour of the vortex embedded in a linear shear flow whose strength changes harmonically in time. Aspects of the vortex dynamics and the (chaotic) transport of tracers have been studied by both laboratory experiments and numerical simulations based on a simple kinematical model.     

The decay of confined vortex rings

Vortex rings are produced during the ejection of fluid through a nozzle or orifice, which occurs in a wide range of biological conditions such as blood flow through the valves of the heart or through arterial constrictions. Confined vortex ring dynamics, such as these, have not been previously studied despite their occurrence within the biological flow conditions mentioned. In this work, we investigate laminar vortex rings using particle image velocimetry and develop a new semi-empirical model for the evolution of vortex ring circulation subject to confinement. Here we introduce a decay parameter ?? which exponentially grows with increasing vortex ring confinement ratio, the ratio of the vortex ring diameter (D _VR) to the confinement diameter (D), with the relationship $\beta=4.38 \exp(9.5D_{\rm VR}/D),$ resulting in a corresponding increase in the rate of vortex ring circulation decay. This work enables the prediction of circulation decay rate based on confinement, which is important to understanding naturally occurring confined vortex ring dynamics.     

Dynamical eigenfunction decomposition of turbulent channel flow

The results of an analysis of low-Reynolds-number turbulent channel flow based on the Karhunen-Loéve(K-L) expansion are presented. The turbulent flow field is generated by a direct numerical simulation of the Navier-Stokes equations at a Reynolds number Re,= 80 (based on the wall shear velocity and channel half-width). The K-L procedure is then applied to determine the eigenvalues and eigenfunctions for this flow. The random coefficients of the K-L expansion are subsequently found by projecting the numerical flow field onto these eigenfunctions. The resulting expansion captures 90% of the turbulent energy with significantly fewer modes than the original trigonometric expansion. The eigenfunctions, which appear either as rolls or shearing motions, posses viscous boundary layers at the walls and are much richer in harmonics than the original basis functions. Chaotic temporal behaviour is observed in all modes and increases for higher-order eigenfunctions. The structure and dynamical behaviour of the eigenmodes are discussed as well as their use in the representation of the turbulent flow.     

Recent developments in background oriented Schlieren methods for rotor blade tip vortex measurements

The compressible blade tip vortex of rotary wings has been the subject of numerous investigations and its importance for the understanding of the helicopter flow field has been clearly emphasised. Due to its great impact on the dynamics of the flow field, the investigation of the tip vortex is directly linked to issues of flow control and aeroacoustic optimisation. However, among velocity field data, additional core density information on the blade tip vortex is desirable with a view to vortex modelling. In this work we describe an airborne background oriented Schlieren system for full-scale helicopter flight tests as well as the first results of the tomographic reconstruction of the compressible vortex core. We report the measurements of both a 0.4 Mach-scaled rotor model of the MBB BO 105 and the corresponding full-scale helicopter in hover flight condition. The tomographic reconstruction of the data allows us to estimate the density and the radius for the viscous core.     

螺旋桨梢涡不稳定性机理与演化模型研究

螺旋桨尾流场的涡流特性是一个基础但又十分复杂的流体力学问题, 它的复杂性源于其蕴含复杂的漩涡系统, 且该漩涡系统会在高速的剪切层流动中不断演化, 其流体动力学行为, 如由稳定态演变为不稳定态的机理以及复杂工况环境中的流动现象, 一直是流体力学领域的难点和备受关注的热点问题. 从工程应用的角度看, 桨后梢涡的演化特性与船舶结构物的宏观特性直接相关, 更好地理解多工况下螺旋桨尾流的动力学特性, 将有助于改善与振动、噪声以及结构问题等相关的推进器性能, 对综合性能优良的下一代螺旋桨的设计和优化有着重要的现实意义. 本文基于延迟分离涡模拟、大涡模拟和无湍流模型模拟方法以及粒子图像测速流场测试分别开展了螺旋桨尾流动力学特性的数值与试验研究, 对螺旋桨尾流不稳定性的触发机理进行了揭示. 基于均匀来流中螺旋桨梢涡的演化机理, 提出了螺旋桨梢涡演化模型. 该模型能够较为准确地模拟螺旋桨梢涡的演化过程, 预测螺旋桨梢涡融合的时间和位置, 对螺旋桨流噪声预报和控制以及性能优良的螺旋桨设计具有重要意义.      

End effects of nominally two-dimensional thin flat plates

Differences in the structure and dynamics of nominally two-dimensional turbulent wakes are investigated experimentally for a thin flat plate, normal to a uniform flow, with two different end conditions: with and without end plates. Both cases are characterized by Karman-like vortex shedding with broadband low frequency unsteadiness. Both wakes evidence a low frequency flapping motion in addition to the slowly drifting base flow common to cylinder wakes. For the case without end plates, an interaction between the drift motion and the vortex formation process is associated with a much stronger modulation of the quasiperiodic vortex shedding amplitude when compared to the case with end plates where a flapping motion is more strongly expressed. These dynamics underlie structural differences in the mean wake and Reynolds stress fields.     

圆柱尾迹涡的三维演化及结构特征

应用无粘涡丝运动学理论和局部诱导近似(LIA)方法,以Lagrange观点跟踪涡丝在背景流场中运动,用数值方法研究了中等Re数(≈10~3)下圆柱分离尾迹中Kármán涡和涡辫区涡丝的三维演化的机制和动力学过程,及其结构特征。背景流场考虑为尾迹时间平均速度流场和Kármán涡街流场。初始展向小扰动为指数形式和谐波形式。结果指出:Kármán涡和涡辫区中的涡丝具有展向不稳定性,形成流向涡量。在尾迹的初期输运过程中,表现出有序的大、小尺度涡结构。并进一步分析了其产生的机理。     

DNS of flows with helical symmetry

Some flows such as the wakes of rotating devices often display helical symmetry. We present an original DNS code for the dynamics of such helically symmetric systems. We show that, by enforcing helical symmetry, the three-dimensional Navier–Stokes equations can be reduced to a two-dimensional unsteady problem. The numerical method is a generalisation of the vorticity/streamfunction formulation in a circular domain, with finite differences in the radial direction and spectral decomposition along the azimuth. When compared to a standard three-dimensional code, this allows us to reach larger Reynolds numbers and to compute quasi-steady patterns. We illustrate the importance of helical pitch by some physical cases: the dynamics of several helical vortices and a quasi-steady vortex flow. We also study the linear dynamics and nonlinear saturation in the Batchelor vortex basic flow, a paradigmatic example of trailing vortex instability. We retrieve the behaviour of inviscid modes and present new results concerning the saturation of viscous centre modes.