On the use of the Weierstrass-Mandelbrot function to describe the fractal component of turbulent velocity

It is shown that the Weierstrass-Mandelbrot function simulates the irregularity in a turbulent velocity record and yields correct forms for the energy and dissipation spectra. In particular, the universal properties of a corresponding multi-fractal function are demonstrated by showing its ability to reproduce and explain turbulent flow spectra measured near the walls of straight and curved channels and in the obstructed space between a pair of disks corotating in an axisymmetric enclosure. The simulation capabilities of the multi-fractal function strongly suggest that turbulence is fractal in the frequency range of the turbulent energy spectrum where the slope of the logarithm of the spectrum, G, is −3 < G < −1. The scale-independent frequency range of the energy spectrum correctly represented by the multi-fractal function includes the isotropic dissipation subrange (−3 < G < −5/3), the inertial subrange (G = −5/3), and the “inner” portion of the anisotropic large-scale subrange (−5/3 < G < −1).     

Asymmetric vortex shedding flow past an inclined flat plate at high incidence

This paper reports an experimental investigation of the vortex shedding wake behind a long flat plate inclined at a small angle of attack to a main flow stream. Detailed velocity fields are obtained with particle-image velocimetry (PIV) at successive phases in a vortex shedding cycle at three angles of attack, α=20°, 25° and 30°, at a Reynolds number Re≈5,300. Coherent patterns and dynamics of the vortices in the wake are revealed by the phase-averaged PIV vectors and derived turbulent properties. A vortex street pattern comprising a train of leading edge vortices alternating with a train of trailing edge vortices is found in the wake. The trailing edge vortex is shed directly from the sharp trailing edge while there are evidences that the formation and shedding of the leading edge vortex involve a more complicated mechanism. The leading edge vortex seems to be shed into the wake from an axial location near the trailing edge. After shedding, the vortices are convected downstream in the wake with a convection speed roughly equal to 0.8 the free-stream velocity. On reaching the same axial location, the trailing edge vortex, as compared to the leading edge vortex, is found to possess a higher peak vorticity level at its centre and induce more intense fluid circulation and Reynolds stresses production around it. It is found that the results at the three angles of attack can be collapsed into similar trends by using the projected plate width as the characteristic length of the flow.     

Spatial resolution and dissipation rate estimation in Taylor--Couette flow for tomographic PIV

This paper assesses the spatial resolution and accuracy of tomographic particle image velocimetry (PIV). In tomographic PIV the number of velocity vectors are of the order of the number of reconstructed particle images, and sometimes even exceeds this number when a high overlap fraction between adjacent interrogations is used. This raises the question of the actual spatial resolution of tomographic PIV in relation to the various flow scales. We use a Taylor--Couette flow of a fluid between two independently rotating cylinders and consider three flow regimes: laminar flow, Taylor vortex flow and fully turbulent flow. The laminar flow has no flow structures, and the measurement results are used to assess the measurement uncertainty and to validate the accuracy of the technique for measurements through the curved wall. In the Taylor vortex flow regime, the flow contains large-scale flow structures that are much larger than the size of the interrogation volumes and are fully resolved. The turbulent flow regime contains a range of flow scales. Measurements in the turbulent flow regime are carried out for a Reynolds number Re between 3,800 and 47,000. We use the measured torque on the cylinders to obtain an independent estimate of the energy dissipation rate and estimate of the Kolmogorov length scale. The data obtained by tomographic PIV are assessed by estimating the dissipation rate and comparing the result against the dissipation rate obtained from the measured torque. The turbulent flow data are evaluated for different sizes of the interrogation volumes and for different overlap ratios between adjacent interrogation locations. The results indicate that the turbulent flow measurements for the lowest Re could be (nearly) fully resolved. At the highest Re only a small fraction of the dissipation rate is resolved, still a reasonable estimate of the total dissipation rate could be obtained by means of using a sub-grid turbulence model. The resolution of tomographic PIV in these measurements is determined by the size of the interrogation volume. We propose a range of vector spacing for fully resolving the turbulent flow scales. It is noted that the use of a high overlap ratio, that is, 75?%, yields a substantial improvement for the estimation of the dissipation rate in comparison with data for 0 and 50?% overlap. This indicates that additional information on small-scale velocity gradients can be obtained by reducing the data spacing.     

PIV measurements of the near-wake behind a sinusoidal cylinder

The three-dimensional near-wake structures behind a sinusoidal cylinder have been investigated using a particle image velocimetry (PIV) measurement technique at Re=3,000. The mean velocity fields and spatial distributions of ensemble-averaged turbulence statistics for flows around the sinusoidal and corresponding smooth cylinders were compared. The near-wake behind the sinusoidal cylinder exhibited pronounced spanwise periodic variations in the flow structure. Well-organized streamwise vortices with alternating positive and negative vorticity were observed along the span of the sinusoidal cylinder. They suppress the formation of the large-scale spanwise vortices and decrease the overall turbulent kinetic energy in the near-wake of the sinusoidal cylinder. The sinusoidal surface geometry significantly modifies the near-wake structure and strongly controls the three-dimensional vortices formed in the near-wake.     

Particle image velocimetry investigation of the flow-field of a 3D turbulent annular swirling decaying flow induced by means of a tangential inlet

Particle image velocimetry (PIV) has been used in order to measure the three mean components and turbulence intensities of the velocity vector in a swirling decaying flow induced by a tangential inlet in an annulus. This kind of flow motion is found to be very complex, exhibiting three-dimensional and non-axisymmetric characteristics coupled with a free decay of the swirling intensity along the flow path, thereby making it difficult to study. A method allowing the measurement of the three components of the velocity flow-field with a standard PIV system with two-dimensional acquisitions, is presented. The evolution of each velocity component between the inlet and the outlet of the annulus is obtained. Furthermore, the PIV technique is extended to the measurement of turbulent characteristics such as turbulent intensities and dimensionless turbulent energy. The main characteristics of the swirling flow are discussed and the swirl number is estimated as a function of the axial distance from the tangential inlet. Received: 6 July 1998/Accepted: 20 March 1999     

An orthogonal-plane PIV technique for the investigations of three-dimensional vortical structures in a turbulent boundary layer flow

An experimental investigation was carried out regarding a three-dimensional topology of a zero-pressure gradient turbulent boundary layer. In this study, the polarization separation technique has been applied to the PIV measurements. Two mutually perpendicular measurement planes have been employed in x–y and x–z planes, respectively. Synchronization between a stereoscopic PIV with another plane PIV system was made toward the detection of such salient features of the coherent structure as the legs and the head of the hairpin vortices. Polarization rotation via a half-waveplate and subsequent particle image separation using polarizer minimized the spurious particle images. The PIV results clearly demonstrate the presence of hairpin-like coherent vortical structures and coincidence between the near-wall quasi-streamwise vortex pair and the legs of the hairpin vortex.     

基于PIV技术分析颗粒在湍流边界层中的行为

采用粒子图像测速技术(particle image velocimetry,PIV)在平板湍流边界层内开展实验研究,对比颗粒相及单相液体的平均速度剖面、湍流强度、雷诺应力等湍流统计量,分析颗粒在湍流边界层中的行为.利用空间多尺度局部平均涡量的概念提取壁湍流发卡涡展向涡头(顺向涡)并统计其数量规律,得到不同法向位置处顺向涡周围流向脉动速度及流线的空间拓扑结构,比较分析顺向涡发展程度及周围的湍流相干结构.结果发现:与清水工况相比,颗粒相湍流边界层的缓冲层变薄、对数律区下移,湍流强度得到增强,雷诺应力在对数律区有所增大;颗粒的流向脉动速度在展向涡周围的分布与清水工况不同,颗粒能够被流体展向涡周围的猝发过程有效传递;颗粒相的顺向涡涡核较大,且随着法向位置的升高逐渐发展完整,涡和条带在流向上拉伸得更长;同时发现在两种工况下,顺向涡的左下方始终存在一个逆向涡,颗粒相逆向涡的形成弱于单相流体;两种工况下的顺向涡数量均随着法向位置的升高而减少,最后逐渐趋于稳定.      

Experimental investigation of the wall dynamics of the A-pillar vortex flow

The influence of the A-pillar vortex on the wall flow of the side window of a car is investigated experimentally using a 30° dihedron model. The measurement of the unsteady pressure at the wall provides a map of the pressure fluctuation intensity, and a spectral analysis is performed to track the dominant frequencies of the wall pressure fluctuations. The wall flow generated by the vortex structure that develops parallel to the side-wall is characterised by means of particle image velocimetry (PIV). Its structure is analysed and compared to cross-sections of the A-pillar vortex in order to identify the different separation and reattachment lines. A comparison of the field of turbulent kinetic energy obtained by PIV with the map of the pressure fluctuations shows a correlation between the structure of the A-pillar vortex and the pressure fluctuations. It is found that the dominant wall pressure fluctuations are located at the secondary separation line, whereas the primary reattachment line does not show any significant pressure variations, that the A-pillar vortex will not naturally break down and that discrete vortices may be associated with the pressure fluctuations.     

Volumetric three-component velocimetry measurements of the turbulent flow around a Rushton turbine

Volumetric three-component velocimetry measurements have been taken of the flow field near a Rushton turbine in a stirred tank reactor. This particular flow field is highly unsteady and three-dimensional, and is characterized by a strong radial jet, large tank-scale ring vortices, and small-scale blade tip vortices. The experimental technique uses a single camera head with three apertures to obtain approximately 15,000 three-dimensional vectors in a cubic volume. These velocity data offer the most comprehensive view to date of this flow field, especially since they are acquired at three Reynolds numbers (15,000, 107,000, and 137,000). Mean velocity fields and turbulent kinetic energy quantities are calculated. The volumetric nature of the data enables tip vortex identification, vortex trajectory analysis, and calculation of vortex strength. Three identification methods for the vortices are compared based on: the calculation of circumferential vorticity; the calculation of local pressure minima via an eigenvalue approach; and the calculation of swirling strength again via an eigenvalue approach. The use of two-dimensional data and three-dimensional data is compared for vortex identification; a ‘swirl strength’ criterion is less sensitive to completeness of the velocity gradient tensor and overall provides clearer identification of the tip vortices. The principal components of the strain rate tensor are also calculated for one Reynolds number case as these measures of stretching and compression have recently been associated with tip vortex characterization. Vortex trajectories and strength compare favorably with those in the literature. No clear dependence of trajectory on Reynolds number is deduced. The visualization of tip vortices up to 140° past blade passage in the highest Reynolds number case is notable and has not previously been shown.     

湍流边界层拟序结构的实验研究

20世纪60年代后, 先后从流动显示发现了快慢斑、猝发、上升流、下扫流和多种涡结构等湍流边界层的拟序结构. 它们对湍流边界层的摩阻、传热传质和湍动能的产生等特性有重要影响. 涡结构是上述拟序结构的核心, 它影响其它拟序结构的发展和演变. 发卡涡通常被认为是基本涡结构. 发卡涡等涡结构的再生, 是湍流边界层拟序结构能够自持续的必要的因素.壁面低速流上升产生猝发, 是湍流边界层湍能的主要来源; 条件采样是测量猝发频率和其它拟序结构出现频率的重要手段. 流动显示对湍流边界层拟序结构作了大量定性观察, 有许多减阻和增加传热率等应用性研究在此基础上发展起来. 80年代后, 出现了测量湍流边界层的瞬时流速矢量场的多热线法和PIV技术, 三维PIV技术可望将来为湍流边界层的实验研究带来重大进展. 本文评述了流动显示法、多热线法和PIV技术的优点和不足之处, 以及它们在对湍流边界层拟序结构的研究中的贡献.      

用PIV测试涡旋波流场的速度和剪应力分布

采用PIV瞬态流场测试技术,对二维槽道中的涡旋波流场不同相位上的速度分布和应力分布进行了测试和计算,本文定量地描述了槽道中涡旋波的形成过程及发展规律。通过调整振荡流的振幅和活塞行程,分析了Re数和Sr数对涡旋波流动的影响,得出了槽道内剪应力的分布状况以及平均剪切应力的周期性变化规律。深入分析了涡旋波流场强化传递现象的本质。     

应用PIV对角区非定常马蹄涡结构的实验研究

利用PIV技术研究了柱体与平板层流边界层角区的非定常流动结构,流动显示和PIV测量均 表明角区存在3种非定常的马蹄涡模态,即绕合模态、脱落-绕合模态以及脱落-耗散模态, 一定$Re$数下主涡脱落后既可能表现为脱落-绕合模态,也可能表现为脱落-耗散模态. 这主 要取决于模型头部形状对涡轴造成的拉伸以及耗散和扩散程度. PIV测量表明,随雷诺数增 加主涡下方从壁面喷发的反向二次涡逐步增大形成强度和尺度较大的``涡舌', 该``涡舌' 将突入整个涡系所在的边界层,最终将主涡与上游涡系隔离并使其从旋涡生成区涡系脱落. 马蹄涡非定常摆动时具有较复杂的奇点形态组合和演化,反映涡轴受到了交替的拉伸和压缩 作用.     

Time-resolved and volumetric PIV measurements of a transitional separation bubble on an SD7003 airfoil

To comprehensively understand the effects of Kelvin–Helmholtz instabilities on a transitional separation bubble on the suction side of an airfoil regarding as to flapping of the bubble and its impact on the airfoil performance, the temporal and spatial structure of the vortices occurring at the downstream end of the separation bubble is investigated. Since the bubble variation leads to a change of the pressure distribution, the investigation of the instantaneous velocity field is essential to understand the details of the overall airfoil performance. This vortex formation in the reattachment region on the upper surface of an SD7003 airfoil is analyzed in detail at different angles of attack. At a Reynolds number Re _c < 100,000 the laminar boundary layer separates at angles of attack >4°. Due to transition processes, turbulent reattachment of the separated shear layer occurs enclosing a locally confined recirculation region. To identify the location of the separation bubble and to describe the dynamics of the reattachment, a time-resolved PIV measurement in a single light-sheet is performed. To elucidate the spatial structure of the flow patterns in the reattachment region in time and space, a stereo scanning PIV set-up is applied. The flow field is recorded in at least ten successive light-sheet planes with two high-speed cameras enclosing a viewing angle of 65° to detect all three velocity components within a light-sheet leading to a time-resolved volumetric measurement due to a high scanning speed. The measurements evidence the development of quasi-periodic vortex structures. The temporal dynamics of the vortex roll-up, initialized by the Kelvin–Helmholtz (KH) instability, is shown as well as the spatial development of the vortex roll-up process. Based on these measurements a model for the evolving vortex structure consisting of the formation of c-shape vortices and their transformation into screwdriver vortices is introduced.     

Numerical study on the turbulent mixing of planar shock-accelerated triangular heavy gases interface

The interaction of a planar shock wave with a triangle-shaped sulfur hexafluoride (\(\mathrm{SF_6}\)) cylinder surrounded by air is numerically studied using a high resolution finite volume method with minimum dispersion and controllable dissipation reconstruction. The vortex dynamics of the Richtmyer–Meshkov instability and the turbulent mixing induced by the Kelvin–Helmholtz instability are discussed. A modified reconstruction model is proposed to predict the circulation for the shock triangular gas–cylinder interaction flow. Several typical stages leading the shock-driven inhomogeneity flow to turbulent mixing transition are demonstrated. Both the decoupled length scales and the broadened inertial range of the turbulent kinetic energy spectrum in late time manifest the turbulent mixing transition for the present case. The analysis of variable-density energy transfer indicates that the flow structures with high wavenumbers inside the Kelvin–Helmholtz vortices can gain energy from the mean flow in total. Consequently, small scale flow structures are generated therein by means of nonlinear interactions. Furthermore, the occasional “pairing” between a vortex and its neighboring vortex will trigger the merging process of vortices and, finally, create a large turbulent mixing zone.     

Characteristics of strongly-forced turbulent jets and non-premixed jet flames

Previous researchers have demonstrated that strong pulsations of the fuel flow rate can significantly reduce the flame length and luminosity of laminar/transitional non-premixed jet flames. The physical mechanisms responsible for these changes are investigated experimentally in acoustically-forced jet flows where the peak velocity fluctuations are up to eight times the mean flow velocity. Both reacting and non-reacting flows were studied and Reynolds numbers, based on the mean flow properties, ranged from 800 to 10,000 (corresponding to peak Reynolds numbers of 1,450–23,000), and forcing frequencies ranged from 290 to 1,140 Hz. Both the first and second organ-pipe resonance modes of the fuel delivery tube were excited to obtain these frequencies. An analysis of the acoustic forcing characteristics within the resonance tube is provided in order to understand the source of the high amplitude forcing. Flow visualization of jets with first resonant forcing confirms the presence of large-scale coherent vortices and strong reverse flow near the exit of the fuel tube. With second-resonant forcing, however, vortices are not emitted from the tube as they are drawn back into the fuel tube before they can fully form. Increased fine-scale turbulence is associated with both resonant cases, but particularly at second resonance. The power spectra of the velocity fluctuations for a resonantly pulsed jet show the presence of an inertial subrange indicating that the flow becomes fully turbulent even for mean-Reynolds-number jets that are nominally laminar. It is shown that these pulsed jet flows exhibit strong similarities to synthetic jets and that the Strouhal number, based on the maximum velocity at the fuel tube exit, is the dominant parameter for scaling these flows. The Strouhal number determines the downstream location where the coherent vortices breakdown, and is found to provide better collapse of flame length data (both current and previous) than other parameters that have been used in the literature.     

Dynamics of hairpin vortices generated by a mixing tab in a channel flow

To better understand mixing by hairpin vortices, time-series particle image velocimetry (PIV) was applied to the wake of a trapezoidal-shaped passive mixing tab mounted at the bottom of a square turbulent channel (Re _h =2,080 based on the tab height). Instantaneous velocity/vorticity fields were obtained in sequences of 10 Hz in the tab wake in the center plane (x–y) and in a plane (x–z) parallel to the wall. Periodically-shed hairpin vortices were clearly identified and seen to rise as they advected downstream. Experimental evidence shows that the vortex-induced ejection of the near-wall viscous fluid to the immediate upstream is important to the dynamics of hairpin vortices. It can increase the strength of the hairpin vortices in the near tab region and cause generation of secondary hairpin vortices further downstream when the hairpin heads are farther away from the wall. Measurements also reveal the existence of a type of new secondary vortice with the opposite-sign spanwise vorticity. The distribution of vortex loci in the x–y plane shows that the hairpin vortices and the reverse vortices are spatially segregated in distinct layers. Turbulence statistics, including mean velocity profiles, Reynolds stresses, and turbulent kinetic energy dissipation rate distributions, were obtained from the PIV data. These statistical quantities clearly reveal imprints of the identified vortex structures and provide insight into mixing effectiveness. Received: 24 February 2000/Accepted: 24 October 2000     

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     

PIV analysis of near-wake behind a sphere at a subcritical Reynolds number

The vortical structure of near-wake behind a sphere is investigated using a PIV technique in a circulating water channel at Re = 11,000. The measured velocity fields show a detailed vortical structure in the recirculation region such as recirculation vortices, reversed velocity zone, and out-of-plane vorticity distribution. The vorticity distribution of the sphere wake shows waviness in cross-sectional planes. The time-averaged turbulent structures are consistent with the visualized flow showing the onset of shear layer instability. The spatial distributions of turbulent intensities provide turbulent statistics for validating numerical predictions.     

Temporal development of turbulent boundary layers with embedded streamwise vortices

The interaction of streamwise vortices with turbulent boundary layer has been investigated using large-eddy simulation. The initial conditions are a pair of counterrotating Oseen vortices with flow between them directed toward the wall (common-flow-down), superimposed on various instantaneous realizations of a turbulent boundary layer. The time development of the vortices and their interaction with the boundary layer are studied by integrating the filtered Navier-Stokes equations in time. The most important effects of the vortices on the boundary layer are the thinning of the boundary layer between vortices (downwash region) and the thickening of the boundary layer in the upwash region. The vortices first move toward the wall as a result of the self-induced velocity, and then apart from each other because of the image vortices due to the solid wall. The Reynolds stress profiles highlight the highly three-dimensional structure of the turbulent boundary layer modified by the vortices. The presence of significant turbulent activity near the vortex center and in the upwash region suggests that localized instability mechanisms in addition to the convection of turbulent energy by the secondary flow are responsible for this effect. High levels of turbulent kinetic energy and secondary stresses in the vicinity of the vortex center are also observed. The numerical results show good agreement with experimental results.This work was supported by the Office of Naval Research under Grant N00014-89-J-1638. Computer time was supplied by the San Diego Supercomputing Center.     

Bubble clustering and trapping in large vortices. Part 2: Time-dependent trapping conditions

In turbulent, periodically excited jets, interactions between bubbles and large coherent vortices are quantitatively studied. Simultaneous, two-phase PIV (particle image velocimetry) and photographic recordings were applied for tracking the large vortices and bubble structures and for investigating trapping phenomena. In order to quantify the interaction between bubbles and the large vortices that are formed in the shear layer, characteristic phase-averaged quantities were determined by PIV. The time-dependent vortex radius, the vorticity at the vortex centre and the time-dependent trapping conditions, obtained from the simulation of the vortex development, were tested against the experimental data.