Point vortex model of the unsteady separated flow past a semi-infinite plate with transverse motion

Two-dimensional unsteady separated flow past a semi-infinite plate with transverse motion is considered. The rolling-up of the separated shear-layer is modelled by a point vortex whose time-dependent circulation is predicted by an unsteady Kutta condition. A power-law starting flow is assumed along with a power law for the transverse motion. The effects of the motion of the plate on the starting vortex circulation and trajectory are presented. A suitable vortex shedding mechanism is introduced and a class of flows involving several vortices is presented. Finally, some possibilities for actively controlling the production of circulation by moving the plate are discussed.     

数值研究平板方舵激波-湍流边界层干扰

数值研究了平板方舵激波-湍流边界层干扰流场。模拟出了分离激波与弓型激波砬撞后形成的“λ”激波结构;消晰地显示了分离区中的旋涡结构,发现流场中会出现二次分离涡,并从理论上分析了流场对称面涡心形态与非定常的关系,得到了涡心为不稳定螺旋点或出现极限环是非定常流动特征的新结论。     

On relative equilibria and integrable dynamics of point vortices in periodic domains

The motion of two point vortices defines an integrable Hamiltonian dynamical system in either singly or doubly periodic domains. The motion of three point vortices in these domains is also integrable when the net circulation is zero. The relative vortex motion in both domains can be reduced to advection of a passive particle by fixed vortices in an equivalent Hamiltonian system. A survey of the solutions for vortex motion in these systems is discussed. Some initial conditions lead to relative equilibria, or vortex configurations that move without change of shape or size. These configurations can be determined as stagnation points in the reduced problem or through explicit solution of the governing equations. These periodic point-vortex systems present a rich collection of interesting solutions despite the few degrees of freedom, and several questions on this subject remain open.     

Strong vortex/boundary layer interactions

This is the second of two papers on the interaction between a longitudinal vortex pair, produced by a delta-wing at angle of attack, and a turbulent boundary layer developing on a flat plate. In the first paper only the outer parts of the vortices entered the boundary layer whereas in this paper the vortices merge with it. In the resultant interaction, the boundary layer between the vortices is kept thin by lateral divergence and a three-dimensional separation line is formed outboard of each vortex. Turbulent, momentum-deficient fluid containing longitudinal vorticity is entrained from the boundary layer along these lines and wrapped around the vortices. As a consequence, the turbulent region of the vortices increases in size and the circulation slowly decreases. It is shown that the flow near the separation line and in the vortices is complicated, and this interaction is expected to be more difficult to calculate than the first. Detailed mean flow and turbulence measurements are reported.     

非对称槽道中涡旋波的特性研究

利用PIV流场显示技术，对振荡流体在非对称槽道中涡旋波的产生、发展和消失的规律进 行了实验研究和分析，测得了涡旋波流场的速度矢量图，阐明了涡旋波流场周期性变化的特 点. 结合涡动力学方程，深入分析并揭示了做周期性运动的流体能在槽道中产生波的特性这 一规律，从中发现：流体周期变化的非定常性和不对称的槽道结构是形成涡旋波流动的主要 因素. 本文对涡旋波流场中各个旋涡的速度分布和涡量进行了测量和计算，分析了涡旋波 强化传质的机理，并研究了Re数对涡旋波流动的影响     

Unsteady flow calculation in a radial flow centrifugal pump with spiral casing

The prediction of the two-dimensional unsteady flow established in a radial flow centrifugal pump is considered. Assuming the fluid incompressible and inviscid, the velocity field is represented by means of source and vorticity surface distributions as well as a set of point vortices. Using this representation, a grid-free (Lagrangian) numerical method is derived based on the coupling of the boundary element and vortex particle methods. In this context the source and vorticity surface distributions are determined through the non-entry boundary condition together with the unsteady Kutta condition. In order to satisfy Kelvin's theorem, vorticity is shed at the trailing edges of the impeller blades. Then the vortex particle method is used to approximate the convection of the free vorticity distribution. Results are given for a pump configuration experimentally tested by Centre Technique des Industries Mécaniques (CETIM). Comparisons between predictions and experimental data show the capability of the proposed method to reproduce the main features of the flow considered.     

Cross-Stream Vorticity Field Induced by Streamwise Vortices in Unbounded and Wall-Bounded Shear Flows

This computational study examines the unsteady cross-stream vorticity structures that form when one or more streamwise vortices are immersed in homogeneous and boundary-layer shear flows. A quasi-two-dimensional limit is considered in which the velocity and vorticity fields, while still possessing three nonzero components, have vanishing gradient in the streamwise direction. This idealization is suitable to applications such as streamwise vortices that occur along a ship hull or airplane fuselage and it can be used as an idealized representation of the quasi-streamwise vortices in the near-wall region of a turbulent boundary layer. In this quasi-two-dimensional idealization, the streamwise velocity has no effect on the cross-stream velocity associated with the vortex. However, the vortex acts to modify the cross-stream vorticity component, resulting in regions of the flow with strong deviations in streamwise velocity. This paper examines the complex structures that form as the cross-stream vorticity field is wrapped up by the vortex and the effect of these structures on the streamwise velocity field, first for vortices immersed in homogeneous shear flow and then for vortices immersed in a boundary layer along a flat wall. Received 2 January 2002 and accepted 13 August 2002 Published online 3 December 2002 RID="*" ID="*" This project was supported by the Office of Naval Research under Grant Number N00014-01-1-0015. Dr. Thomas Swain is the program manager. Communicated by T.B. Gatski     

The effect of wall cooling on compressible Görtler vortices

It is known that in adiabatic boundary layer flow over a curved surface the detailed structure of the spanwise periodic Görtler vortex instability varies markedly over the range of spanwise wavelength. At short wavelengths the modes tend to be concentrated in a well-defined thin zone located within the boundary layer. As the vortex wavenumber diminishes so the region of vortex activity is first driven to the bounding wall but subsequently expands to cover the entire boundary layer at which stage the modes take on a principally inviscid form. At yet longer wavelengths the vortices are given by the solution of an interactive multi-deck structure which has some similarities with that for Tollmien–Schlichting waves.In this work we investigate how the application of wall cooling affects the above scenario. It is shown how cooling both restricts the range of mode types and gives rise to two new structures. The first, for moderate cooling and which relates to longer wavelengths, is interactive in nature. Here the viscous–inviscid interaction between an essentially inviscid Görtler problem, albeit for an effective basic flow which in its general form has a non-standard near-wall structure, and a viscous sublayer is provided by novel boundary conditions. Shorter wavelength vortices are largely unaffected by wall cooling unless this is quite severe. However when this degree of cooling is applied, the vortices take on a fully viscous form and are confined to a thin region next to the bounding wall wherein the basic flow assumes an analytic form. Numerical solutions are obtained and we provide evidence as to how the two new structures are related both to each other and to the previously known uncooled results.     

Flow structure in a flat vortex chamber

We investigate a flow in a flat vortex chamber in which the distance between the end walls is smaller than the radius of the chamber. The study was mainly performed by optical methods: a Töpler device was employed, with the Foucault knife replaced by a diaphragm. It is shown that the flow in the chamber has a complicated spatial structure. In addition to the basic helical flow, an intense “transverse” rotation of the type of Taylor-Görtler vortices occurs. In contrast to previously studied flows, where these vortices were observed near a concave surface, in the motion considered transverse vortices occur in the entire working volume of the chamber. In this case, four parallel vortex filaments are formed. The high intensity of the vortices has allowed one to visualize them by the Töpler method and by “tinting” the flow by highly disperse particles. Quantitative dependences of the dimensions of the vortex cells on the flow regime, i.e., on the pressure of gas deceleration, were obtained.     

Three-dimensional instability analysis of boundary layers perturbed by streamwise vortices

A parametric study is presented for the incompressible, zero-pressure-gradient flat-plate boundary layer perturbed by streamwise vortices. The vortices are placed near the leading edge and model the vortices induced by miniature vortex generators (MVGs), which consist in a spanwise-periodic array of small winglet pairs. The introduction of MVGs has been experimentally proved to be a successful passive flow control strategy for delaying laminar-turbulent transition caused by Tollmien–Schlichting (TS) waves. The counter-rotating vortex pairs induce non-modal, transient growth that leads to a streaky boundary layer flow. The initial intensity of the vortices and their wall-normal distances to the plate wall are varied with the aim of finding the most effective location for streak generation and the effect on the instability characteristics of the perturbed flow. The study includes the solution of the three-dimensional, stationary, streaky boundary layer flows by using the boundary region equations, and the three-dimensional instability analysis of the resulting basic flows by using the plane-marching parabolized stability equations. Depending on the initial circulation and positioning of the vortices, planar TS waves are stabilized by the presence of the streaks, resulting in a reduction in the region of instability and shrink of the neutral stability curve. For a fixed maximum streak amplitude below the threshold for secondary instability (SI), the most effective wall-normal distance for the formation of the streaks is found to also offer the most stabilization of TS waves. By setting a maximum streak amplitude above the threshold for SI, sinuous shear layer modes become unstable, as well as another instability mode that is amplified in a narrow region near the vortex inlet position.     

基于涡旋识别方法的高速列车尾涡结构的讨论

利用改进型延迟分离涡模拟方法对缩尺比例1:30的高速列车简化模型的绕流流场进行数值计算,主要针对近尾流区的涡旋结构展开具体讨论. 通过不同的涡旋识别方法,发现在尾涡结构中,高涡量的强涡旋主要聚集于尾车附近,而涡量较低但处于相对稳定状态的涡旋分布在大部分尾流空间中. 对此,主要基于最新提出的涡旋定义及其物理意义认为,由于边界层在尾部发生的流动分离,剪切变形以及高涡量的扩散对强涡旋的形成发挥着重要的作用,而涡旋会被较强的剪切旋转拉伸,使得局部复杂的流动表现出突出的湍流特性;另一方面,尽管涡强度明显下降,但是在强剪切应变迅速衰减的情况下,流向涡核中的涡旋涡量是主要的,此时,在较接近地面的情况下,流体微团以涡核为中心的旋转运动使得涡旋与地面之间的相互作用成为主导的流动机制. 虽然涡强度会相对缓慢地衰减,但是从湍流能量产生的角度,该机制对涡旋的自维持发挥重要的作用,从而使尾涡结构能够相对稳定地存在于尾流流动中.      

Point vortex dynamics: recent results and open problems

he concept of point vortex motion, a classical model in the theory of two-dimensional, incompressible fluid mechanics, was introduced by Helmholtz in 1858. Exploration of the solutions to these equations has made fitful progress since that time as the point vortex model has been brought to bear on various physical situations: atomic structure, large scale weather patterns, “vortex street” wakes, vortex lattices in superfluids and superconductors, etc. The point vortex equations also provide an interesting example of transition to chaotic behavior. We give a brief historical introduction to these topics and develop two of them in particular to the point of current understanding: (i) Steadily moving configurations of point vortices; and (ii) Collision dynamics of vortex pairs.     

The effect of fine structure on the stability of planar vortices

This study considers the linear, inviscid response to an external strain field of classes of planar vortices. The case of a Gaussian vortex has been considered elsewhere, and an enstrophy rebound phenomenon was noted: after the vortex is disturbed enstrophy feeds from the non-axisymmetric to mean flow. At the same time an irreversible spiral wind-up of vorticity fluctuations takes place. A top-hat or Rankine vortex, on the other hand, can support a non-decaying normal mode.In vortex dynamics processes such as stripping and collisions generate vortices with sharp edges and often with bands or rings of fine scale vorticity at their periphery, rather than smooth profiles. This paper considers the stability and response of a family of vortices that vary from a broad profile to a top-hat vortex. As the edge of the vortex becomes sharper, a quasi-mode emerges and vorticity winds up in a critical layer, at the radius where the angular velocity of the fluid matches that of a normal mode on a top-hat vortex. The decay rate of these quasi-modes is proportional to the vorticity gradient at the critical layer, in agreement with theory. As the vortex edge becomes sharper it is found that the rebound of enstrophy becomes stronger but slower.The stability and linear behaviour of coherent vortices is then studied for distributions which exhibit additional fine structure within the critical layer. In particular we consider vorticity profiles with ‘bumps’, ‘troughs’ or ‘steps’ as this fine structure. The modified evolution equation that governs the critical layer is studied using numerical simulations and asymptotic analysis. It is shown that depending on the form of the short-scale vorticity distribution, this can stabilise or destabilise quasi-modes, and it may also lead to oscillatory behaviour.     

An unsteady point vortex method for coupled fluid–solid problems

A method is proposed for the study of the two-dimensional coupled motion of a general sharp-edged solid body and a surrounding inviscid flow. The formation of vorticity at the body’s edges is accounted for by the shedding at each corner of point vortices whose intensity is adjusted at each time step to satisfy the regularity condition on the flow at the generating corner. The irreversible nature of vortex shedding is included in the model by requiring the vortices’ intensity to vary monotonically in time. A conservation of linear momentum argument is provided for the equation of motion of these point vortices (Brown–Michael equation). The forces and torques applied on the solid body are computed as explicit functions of the solid body velocity and the vortices’ position and intensity, thereby providing an explicit formulation of the vortex–solid coupled problem as a set of non-linear ordinary differential equations. The example of a falling card in a fluid initially at rest is then studied using this method. The stability of broadside-on fall is analysed and the shedding of vorticity from both plate edges is shown to destabilize this position, consistent with experimental studies and numerical simulations of this problem. The reduced-order representation of the fluid motion in terms of point vortices is used to understand the physical origin of this destabilization.      

VORTEX SHEDDING OF BLUFF BODIES: A REVIEW

In this paper the various types of vortex generation and the related response characteristics of bluff bodies are described. The vortices are, in general, generated by a certain stimulation, leading to one- or two-shear layer instability; the related unsteady forces could excite flexible structures such as tall towers, tall buildings and long-span bridges. Karman vortex shedding is well known as the alternate shedding vortex behind bluff bodies, but the one-shear layer instability related vortices and symmetrical vortex shedding should also be taken into account as additional mechanisms for the evaluation of structural safety, because they result in structural response at comparatively low wind speeds. In this paper, the symmetrical vortex shedding, which is enhanced by the longitudinally fluctuating flow for 2-D rectangular cylinders with a 0.5 side ratio, and one-shear layer related vortices, which are generated on the side surfaces of flat 2-D rectangular cylinders and many bridge girder box sections by the stimulation of body motion or applied sound, are introduced. Furthermore, as a peculiar 3-D vortex, the “axial vortex”, which is formed in near wake of inclined cables and then over restricted velocity ranges, is also discussed.     

Numerical studies on evolution of secondary streamwise vortices in transitional boundary layers

Formation and evolution of secondary streamwise vortices in the compressible transitional boundary layers over a flat plate are studied using a direct numerical simulation method with high-order accuracy and highly effective non-reflecting characteristic boundary conditions. Generation and development processes of the secondary streamwise vortices in the complicated transitional boundary flow are clearly analyzed based on the of numerical results, and the effects on the formation of the ring-like vortex that is vital to the boundary layer transition are explored. A new mechanism forming the ring-like vortex through the mutual effect of the primary and secondary streamwise vortices is expressed.     

Analytical model of motion of turbulent vortex rings in an incompressible fluid

An analytical model describing the motion of vortex rings in an incompressible fluid is constructed. The model is valid both for homogeneous and inhomogeneous vortices buoyant in the gravity field, as well as for combined vortices. The expansion angle of a buoyant vortex is found from the characteristic parameters that define the flow rather than specified on the basis of experiments. Significant differences in the expansion angles of homogeneous and buoyant vortex rings are explained. The calculation results for the proposed model are compared with the results of laboratory experiments and data on the rise of the cloud produced by an atomic explosion.     

Coaxial interactions of two vortex rings or of a ring with a body

Inviscid coaxial interactions of two vortex rings, including head-on collisions and leapfrogging motions, are considered using a contour dynamics technique. Interactions of vortex rings with solid bodies are also investigated by combining the contour dynamics technique with a boundary integral equation method. Numerical results show that a clean, successful passage motion is possible for two vortex rings with not too thick cores. In both cases of head-on collisions and leapfrogging motions, very large core deformations are observed when two vortex rings get close to each other. A head-tail structure is formed in the later stage of a head-on collision of two fat vortices. Numerical results also show that a vortex ring will stretch and slow down when it moves toward a solid boundary, will shrink and speed up when it moves away from a solid boundary, and will either translate steadily or approach an oscillating asymptotic state when it is far away from any boundaries. The project supported by The National Education Commission of China and NASA under cooperative grant agreement #NCC5-34.     

Study on multiple ring-like vortex formation and small vortex generation in late flow transition on a flat plate

A high-order direct numerical simulation (DNS) of flow transition over a flat plate at a free stream Mach number 0.5 has been carried out. During the simulation, we cannot find, according to the classical theory of boundary layer transition, the “hairpin vortex breakdown to smaller structures” in the last stage of flow transition on a flat plate. However, we did discover the so-called spikes as a result of a multibridge or multiring formation. This indicated a large and stable vortex structure which can travel for a long distance. We believe that this is a result of ring heads that are located in an inviscid region. These heads of the “turbulence spot” never seem to break down and persist as a stable structure. In addition, we discovered that the U-shaped vortex is a part of an existing coherent structure instead of a newly generated one. The U-shaped vortex also provides an additional channel to transfer vorticity to the ring from the wall. During travel, the leading primary ring in the front of the spot is sloped and skewed, causing disappearance of the second sweep. As a consequence, no energy is brought down to the lower boundary layer near the vortex ring head of the spot. Thus, the small length scale structures become damped and the existing U-shaped vortex structure becomes distinguishable. So, the question is where do the turbulent small length scale vortices come from? We will address this with a new theory which states that all of small length scales (turbulence) are generated by high-shear (HS) layers rather than being produced by “vortex breakdown.” The new DNS shows that the HS layers are produced by strong positive spikes surrounded by low-speed fluids and by the interaction between the secondary and higher-level vortices and the wall surface especially near the ring neck. This multiple ring-like vortex generation also follows the first Helmholtz vortex conservation law. Furthermore, the Λ-shaped vortex is formed and rolling up, and the Λ-vortex is stretched and narrowed, and a new bridge is generated after the neck. The bridge will further become a second ring and so on. Besides the original vortex legs, there are also U-shaped vortex tubes. Finally, the multiple ring vortex structure is formed. From this process, no evidence is found to support that two consequent multiring circles are mixed to generate small vortices. The connection of downdraft/updraft motions is also studied.     

Surface pressure on a cubic building exerted by conical vortices

Experiments were performed to study surface pressure on a cubic building underlying conical vortices, which are known to cause severe structural damage and failure. The focus is on the effects of turbulence in the incident flow. Three turbulent boundary layers were created in a boundary layer wind tunnel. A wall-mounted cube, i.e. a cube situated on the horizontal ground floor surface of the wind-tunnel test section, was used as an experimental model. The cube was subjected to the incidence flow at 40°. Steady and unsteady pressure measurements were performed on the cube surface. The analysis suggests that conical vortices developed above the top surface of the wall-mounted cube. A larger mean suction was observed on the top cube surface in the less turbulent boundary layer. With an increase in turbulence in the incoming flow, the strong suction zones decreased in size. The fluctuating pressure coefficient profiles retained their shape when the turbulence in the upstream flow of the cube increased. The fluctuating pressure coefficient was observed to be larger in more turbulent flows. The pressure fluctuations were larger on the cube surface underlying outer boundaries of the conical vortex. The fluctuating pressure coefficient under the conical vortex was three to four times larger than in the weak suction zone on the central area of the top cube surface. Close to the leading cube corner, the pressure spectra were dominated by a single low frequency peak. As the conical vortex developed, this primary peak weakened and a secondary peak emerged at a higher reduced frequency. There is a general trend of shifting the pressure spectra towards higher reduced frequencies when the turbulence in the undisturbed incident flow increases.