Experimental characterization of the instability of the vortex rings. Part II: Non-linear phase

The first stage of the instability of a vortex ring is linear and characterized by the growth of an azimuthal stationary wave which develops around the ring. Theoretical works predict its origin, shape, number of waves and growth rate. Apart for the growth rate, experimental and numerical results in viscous fluids fit well with the predictions based on an ideal fluid hypothesis. On the other hand, the next stages of the development of the instability (which are non-linear) are not well known. Only few phenomena are described, in an isolated way, in various partial contributions. The aim of this paper is to report on a complete experimental investigation of the non-linear phase of the instability of the vortex ring. The vortices were produced in water and their Reynolds number Re _p was varied from 2,650 to 6,100. Visualizations were performed using planar laser induced fluorescence and measurements with 2D2C and 2D3C particle image velocimetry. Based on a Fourier analysis of the results, it appears that the non-linear phase begins with the development of harmonics of the linear modes (first unstable modes). But the growth of those harmonics is rapidly stopped by the development of low order modes. Then appears an m=0 mode, which corresponds to a mean azimuthal velocity around the vortex. Simultaneously, secondary vortical structures develop all around the vortex in its peripheral zone. These vortical structures are linked with the ejection of vorticity in the wake of the ring and they appear just before the transition towards turbulence. A tentative is made here to place all these phenomena chronologically, in order to propose a scenario for the transition from the linear phase to turbulence.     

Instability of a planar liquid sheet with surrounding fluids between two parallel walls

We investigate the linear and nonlinear instability of a planar liquid sheet with surrounding fluids between two parallel plane solid walls. Linear analysis shows that the maximum temporal growth rate and unstable wave number region of disturbances increase for the dilational and sinuous modes when the gap between the sheet and the wall decreases. The walls have more influence on the instability when the density ratio of the surrounding fluid to the sheet and/or the Weber number decrease. On the other hand, nonlinear analysis is performed by means of the discrete vortex method, where double vortex rows and their mirror images are placed so as to satisfy the boundary condition on the walls. Numerical results show that the walls enhance nonlinearity, which causes deformation and distortion of the sheet, whereas the nonlinearity diminishes linear growth rates except for long dilational disturbances. In particular, as the walls are placed more closely to the sheet, local sheet thinning becomes more pronounced in the long dilational mode, while the dilational mode is more strongly induced from the sinuous mode through monotonic or periodic energy exchanges between the two modes.     

Linear instability in the wake of an elliptic wing

Linear global instability analysis has been performed in the wake of a low aspect ratio three-dimensional wing of elliptic cross section, constructed with appropriately scaled Eppler E387 airfoils. The flow field over the airfoil and in its wake has been computed by full three-dimensional direct numerical simulation at a chord Reynolds number of \(Re_{c}=1750\) and two angles of attack, \(\mathrm{{AoA}}=0^\circ \) and \(5^\circ \). Point-vortex methods have been employed to predict the inviscid counterpart of this flow. The spatial BiGlobal eigenvalue problem governing linear small-amplitude perturbations superposed upon the viscous three-dimensional wake has been solved at several axial locations, and results were used to initialize linear PSE-3D analyses without any simplifying assumptions regarding the form of the trailing vortex system, other than weak dependence of all flow quantities on the axial spatial direction. Two classes of linearly unstable perturbations were identified, namely stronger-amplified symmetric modes and weaker-amplified antisymmetric disturbances, both peaking at the vortex sheet which connects the trailing vortices. The amplitude functions of both classes of modes were documented, and their characteristics were compared with those delivered by local linear stability analysis in the wake near the symmetry plane and in the vicinity of the vortex core. While all linear instability analysis approaches employed have delivered qualitatively consistent predictions, only PSE-3D is free from assumptions regarding the underlying base flow and should thus be employed to obtain quantitative information on amplification rates and amplitude functions in this class of configurations.     

Inviscid instabilities in two-dimensional spatially periodic flows

The instability of two-dimensional vortex arrays with hexagonal symmetry is investigated with respect to inviscid perturbations. A numerical treatment in the framework of Floquet theory provides critical parameters for instability onset and the spatial structure of unstable modes which are in agreement with experiments in electromagnetically driven flows conducting fluids. The stability of point vortex lattices is briefly discussed.     

Temporal and spatial instability of an inviscid compound jet

This paper examines the linear hydrodynamic stability of an inviscid compound jet. We perform the temporal and the spatial analyses in a unified framework in terms of transforms. The two analyses agree in the limit of large jet velocity. The dispersion equation is explicit in the growth rate, affording an analytical solution. In the temporal analysis, there are two growing modes, stretching and squeezing. Thin film asymptotic expressions provide insight into the instability mechanism. The spatial analysis shows that the compound jet is absolutely unstable for small jet velocities and admits a convectively growing instability for larger velocities. We study the effect of the system parameters on the temporal growth rate and that of the jet velocity on the spatial growth rate. Predictions of both the temporal and the spatial theories compare well with experiment.Dedicated to the memory of Professor Tasos C. Papanastasiou     

Impact of buoyancy on vortex ring development in the near field

The development of a buoyant vortex ring in the near field was examined experimentally, and the findings were compared with those of a non-buoyant ring with a similar Reynolds number. The experiments were performed in a water tank, and the vortices were produced by a cylindrical tube of aspect ratio 2. Laser sheet flow visualization and PIV measurements were carried out. In the near field, the initial column of the buoyant fluid breaks down due to the presence of Rayleigh–Taylor instability at the buoyant fluid interface. Subsequently, a large diameter vortex ring with a large spreading rate, compared with the non-buoyant ring, emerges. The celerity of buoyant vortex continued to decrease throughout the range examined, in contrast to the constant celerity of the non-buoyant ring. The vorticity in the core of buoyant and non-buoyant vortex rings is symmetric and has a Gaussian distribution. However, the buoyant vortex ring evolves into a thin core ring, whereas the non-buoyant ring becomes a thick core ring shortly after the ring formation. This difference is brought on by the rapid entrainment and the significant growth of the buoyant ring following the breakup of the initial formation.     

Hierarchical instability of a vortex ring array in multipulse laser-matter interactions

In XeCl excimer Laser interactions with Co-coated steel surfaces, we have seemingly realized a quasi-linear array of vortex rings. These form from instabilities on an array of vortex filaments which emerge and then lead to a series of loops. Finally the collapse-and-reconnection process yields a cascade of nearby vortex rings. Since the filament array is subjected to randomly distributed local multipolar strains, unstable waves can develop on the vortex rings. These deformed shapes are frozen permanently by ultrafast cooling, following the last laser pulse. Using a modal analysis, an attempt is made to relate the wave structures to a parametric resonance instability which is caused by dipolar and a quadrupolar fields. Multipulse laser-matter interactions are thus capable of nonselective excitation of vortex ring instabilities of various sizes and various modal structures.     

Development of the instability of an axisymmetric vortex flow in a circular cylinder

A vortex structure consisting of two concentric vortex rings located in a circular cylinder is studied. The rings touch each other and have a uniform vorticity. The geometric and dynamic parameters of the rings satisfy the condition of zero total intensity. A linear stability analysis of this structure is performed, including the limiting cases when the inner ring transforms into a circle or the outer ring is adjacent to the boundary of the cylinder. The results of numerical simulation of the evolution of unsteady flows are presented for a wide range of variation of governing parameters of the problem. It is established that the evolution of unstable vortex flows results in the formation of several types of quasistationary structures, the topology of which depends mainly on the dominant disturbance mode.     

Instability waves and low-frequency noise radiation in the subsonic chevron jet

Spatial instability waves associated with lowfrequency noise radiation at shallow polar angles in the chevron jet are investigated and are compared to the round counterpart. The Reynolds-averaged Navier–Stokes equations are solved to obtain the mean flow fields, which serve as the baseflow for linear stability analysis. The chevron jet has more complicated instability waves than the round jet, where three types of instability modes are identified in the vicinity of the nozzle, corresponding to radial shear, azimuthal shear,and their integrated effect of the baseflow, respectively. The most unstable frequency of all chevron modes and round modes in both jets decrease as the axial location moves downstream. Besides, the azimuthal shear effect related modes are more unstable than radial shear effect related modes at low frequencies. Compared to a round jet, a chevron jet reduces the growth rate of the most unstable modes at downstream locations. Moreover, linearized Euler equations are employed to obtain the beam pattern of pressure generated by spatially evolving instability waves at a dominant low frequency St = 0.3, and the acoustic efficiencies of these linear wavepackets are evaluated for both jets. It is found that the acoustic efficiency of linear wavepacket is able to be reduced greatly in the chevron jet, compared to the round jet.     

Three-dimensional receptivity of boundary layers

The paper presents a review of results of some recent (mainly experimental) studies devoted to a quantitative investigation of the problem of receptivity of the 2D and 3D boundary layers with respect to various 3D (in general) external perturbations. The paper concentrates on the mechanisms of excitation and development of stationary and travelling instability modes in a 3D boundary layer on a swept wing, as well as in 2D boundary layers including the Blasius flow and a self-similar boundary layer with an adverse pressure gradient. In particular, the following problems of the boundary-layer receptivity are discussed: (i) receptivity to localized 3D surface roughness, (ii) receptivity to localized 3D surface vibrations, (iii) acoustic receptivity in presence of 3D surface roughness, and (iv) acoustic receptivity in the presence of 3D surface vibrations. All experiments described in the paper were conducted using controlled disturbance conditions with the help of simulation of the stationary and non-stationary perturbations by means of several disturbance generators. This approach gives us the possibility to obtain quantitative results which are independent of any uncontrolled background perturbations of the flow and the experimental model. In contrast to the data obtained at “natural” environmental conditions these results can be directly compared with calculations without any significant assumptions about the physical nature of the disturbances under investigation. The complex (amplitude and phase) coefficients of the boundary-layer receptivity to external perturbations, obtained as functions of the disturbance frequency and the spanwise wavenumber (or the wave propagation angle), represent the main results of the experiments described. These results can be used for the evaluation of the initial amplitudes and phases of the instability modes generated by various external perturbations, as well as for quantitative verification of linear receptivity theories. Several examples of the comparison of experimental results with calculations are also presented in this paper. A brief analysis of the state-of-art in the field is performed and some general properties of different receptivity mechanisms are discussed.     

Penetration length and diameter development of vortex rings generated by impacting water drops

The formation of water drops underneath nozzles was studied numerically to provide the basis for repeatable experimental drop formation. The drops detached from the nozzle and impacted on a free water surface forming vortex rings. Experimental results obtained through kinematographic studies of the penetration and growth of these vortex rings are presented. Variations of the penetration depth were measured, depending on the nozzle height above the water surface where the drops were formed. The experiments revealed that it is the state of oscillation at impact that defines the penetration length. Interesting variations of the vortex ring diameter with time were recorded. A good overall understanding of the behavior of the diameter variation was obtained when it was plotted as a function of the penetration depth.     

Obstacle-induced spiral vortex breakdown

An experimental investigation on vortex breakdown dynamics is performed. An adverse pressure gradient is created along the axis of a wing-tip vortex by introducing a sphere downstream of an elliptical hydrofoil. The instrumentation involves high-speed visualizations with air bubbles used as tracers and 2D Laser Doppler Velocimeter (LDV). Two key parameters are identified and varied to control the onset of vortex breakdown: the swirl number, defined as the maximum azimuthal velocity divided by the free-stream velocity, and the adverse pressure gradient. They were controlled through the incidence angle of the elliptical hydrofoil, the free-stream velocity and the sphere diameter. A single helical breakdown of the vortex was systematically observed over a wide range of experimental parameters. The helical breakdown coiled around the sphere in the direction opposite to the vortex but rotated along the vortex direction. We have observed that the location of vortex breakdown moved upstream as the swirl number or the sphere diameter was increased. LDV measurements were corrected using a reconstruction procedure taking into account the so-called vortex wandering and the size of the LDV measurement volume. This allows us to investigate the spatio-temporal linear stability properties of the flow and demonstrate that the flow transition from columnar to single helical shape is due to a transition from convective to absolute instability.     

Explosive instability of geostrophic vortices. Part 1: baroclinic instability

In a quasi-geostrophic model, we study the baroclinic instability of a two-layer vortex. The singular unstable modes for potential vorticity anomalies are compared with the classical normal modes. Short-time singular modes are explosively unstable and, at short times, depend only on the baroclinic component of the flow. As time progresses, they evolve towards the normal modes and their sensitivity to flow parameters is explored. Asymptotic solutions are provided.     

气流作用下同轴带电射流的不稳定性研究

通过对气体驱动同轴电流动聚焦的实验模型进行简化,开展了电场力和惯性力共同作用下同轴带电射流的不稳定性理论研究.在流动为无黏、不可压缩、无旋的假设下,建立了三层流体带电射流物理模型并得到了扰动在时间域内发展演化的解析形式色散关系,利用正则模方法求解色散方程发现了流动的不稳定模态,进而分析了主要控制参数对不稳定模态的影响.结果表明,在参考状态下轴对称模态的最不稳定增长率最大,因此轴对称扰动控制整个流场.外层气流速度越高,气体惯性力越大,射流的界面越容易失稳.内外层液-液同轴射流之间的速度差越大,射流越不稳定.表面张力对射流不稳定性起到促进作用.轴向电场对射流不稳定性具有双重影响:当加载电场强度较小时,射流不稳定性被抑制;当施加电压大于某一临界值时,轴向电场会促进射流失稳.临界电压的大小与界面上自由电荷密度和射流表面扰动发展关系密切.这些结果与已有的实验现象吻合,能够对实验的过程控制提供理论指导.     

Linear stability of swirling flows computed as solutions to the quasi-cylindrical equations of motion

The linear stability of numerical solutions to the quasi-cylindrical equations of motion for swirling flows is investigated. Initial conditions are derived from Batchelor's similarity solution for a trailing line vortex. The stability calculations are performed using a second-order-accurate finite-difference scheme on a staggered grid, with the accuracy of the computed eigenvalues enhanced through Richardson extrapolation. The streamwise development of both viscous and inviscid instability modes is presented. The possible relationship to vortex breakdown is discussed.     

Evolution of a forced counter rotating vortex pair for two selected forcing frequencies

In this paper, results from an experimental study of the natural and forced evolution of a pair of counter rotating wing-tip vortices are reported. The vortices were generated using a pair of opposed wing-tips in a wind tunnel and measurements made up to 77 tip-spacings downstream of the models at a chord Reynolds number of 1.3 × 10⁵. The wake was interrogated using 2D particle image velocimetry and the long-wave Crow instability observed. Velocity data were recorded throughout the lifetime of the instability from initial growth through linking, formation of vortex rings and their subsequent decay. Forcing was achieved using pulsed air jets blowing in the span-wise direction from the wing tip and imparting spatially periodic perturbations to the vortices. Forcing at a frequency within the range amplified by the Crow instability was found to enhance the instability growth rate whereas forcing at a frequency outside the amplified range was found to inhibit instability growth. In the latter case the imparted wavelength was observed to die out with a preferred wavelength growing in its place.     

Power-Spectral density estimate of the Bloor-Gerrard instability in flows around circular cylinders

There have been differences in the literature concerning the power law relationship between the Bloor-Gerrard instability frequency of the separated shear layer from the circular cylinder, the Bénard-von Kármán vortex shedding frequency and the Reynolds number. Most previous experiments have shown a significant degree of scatter in the measurement of the development of the shear layer vortices. Shear layers are known to be sensitive to external influences, which can provide a by-pass transition to saturated growth, thereby camouflaging the fastest growing linear modes. Here, the spatial amplification rates of the shear layer instabilities are calculated using power-spectral density estimates, allowing the fastest growing modes rather than necessarily the largest structures to be determined. This method is found to be robust in determining the fastest growing modes, producing results consistent with the low scatter results of previous experiments.     

Numerical simulation of three-dimensional instability of flow past a short cylinder

The plane-parallel flow past an infinitely long circular cylinder becomes three-dimensional starting with Reynolds numbers Re ≈ 190. The corresponding instability mode is called mode A. When Re ≈ 260, vortex structures with a smaller cross scale are formed in the wake as a result of a secondary three-dimensional instability (mode B). The transition to three-dimensionality for a short cylinder bounded by planes is considered. The length of the cylinder is chosen to eliminate the unstable perturbations of mode A. Two instability modes similar to modes A and B modified under the effect of the bounding lateral planes are found. The problems of three-dimensional flow are numerically solved using the Navier-Stokes equations.     

Phase-locking in a jet forced with two frequencies

This study investigates the effect of forcing a shear layer at more than one frequency. Multiple frequency forcing permits the phase and initial relative amplitudes among unstable waves to be manipulated. More control can be imposed on vortex merging and mixing. Various vortex merging modes were observed and explained by the relative strength of the instability waves and their phase alignment. The vortex phase and path jitterings present in single-frequency forcing cases are greatly reduced when forced at more than one frequency. The observed cycle-to-cycle variation was small. This enables phase-lock measurements to be performed more easily. The phase-lock data show excellent agreement with the flow visualization results even when averaged over only a few cycles.     

On Tollmien–Schlichting-like waves in streaky boundary layers

The linear stability of the boundary layer developing on a flat plate in the presence of finite-amplitude, steady and spanwise periodic streamwise streaks is investigated. The streak amplitudes considered here are below the threshold for onset of the inviscid inflectional instability of sinuous perturbations. It is found that, as the amplitude of the streaks is increased, the most unstable viscous waves evolve from two-dimensional Tollmien–Schlichting waves into three-dimensional varicose fundamental modes which compare well with early experimental findings. The analysis of the growth rates of these modes confirms the stabilising effect of the streaks on the viscous instability and that this stabilising effect increases with the streak amplitude. Varicose subharmonic modes are also found to be unstable but they have growth rates which typically are an order of magnitude lower than those of fundamental modes. The perturbation kinetic energy production associated with the spanwise shear of the streaky flow is found to play an essential role in the observed stabilisation. The possible relevance of the streak stabilising role for applications in boundary layer transition delay is discussed.