Note on the POD-based time interpolation from successive PIV images

To enhance the temporal resolution of the PIV measurements of pseudo-periodic turbulent flows, Proper Orthogonal Decomposition (POD) has been previously used to time interpolate PIV database. In this note, it is demonstrated that such POD interpolation is equivalent to the classical mathematical interpolation when dealing with the whole POD eigenfunctions, since POD is a linear transform. In fact, the POD-based time interpolation is only valid for the large scale structures of the flow. The advantage of using POD procedure resides in its efficiency in extracting the dominant flow structures. In this sense, other interpolation methods such as turbulent filtering procedures could provide similar results. To cite this article: E. Bouhoubeiny, P. Druault, C. R. Mecanique 337 (2009).     

Extension of Heisenberg's model for spectral transfer to second order fluids in turbulent flow

A generalization of the eddy viscosity concept permits an extension of the Heisenberg model for spectral energy transfer. The results are applied to the isotropic homogeneous turbulent flow of an incompressible second order fluid. This work was supported by the U.S. Office of Naval Research, Fluid Mechanics Branch, under Contract Nonr 656(33) and is taken from the Ph.D. Dissertation of the first author in the department of Aerospace Engineering, March 1966.     

Distribution and dispersion in the Euler-Lagrange random walk

Summary In a new kind of random walk, displaying both Lagrangian and Eulerian statistical properties¹⁾, calculations were made previously of the Eulerian and Lagrangian velocity auto-correlations. Now the characteristic function of particle displacement has been calculated, and possible continuum limit forms for the probability density equation have been deduced. One of these turns out to be the telegraph equation, given by Goldstein²⁾ as the limit of a different kind of random walk. For this case the auto-correlation functions have been determined.This work was supported by the Mechanics Branch, U.S. Office of Naval Research, under contract Nonr 248 (38) and was presented in Session F of the 1959 Annual Meeting of the American Physical Society at New York under the title A Random Walk with both Lagrangian and Eulerian Statistics.Associate Professor of Engineering Research, Ordnance Research Laboratory, The Pennsylvania State University. This work was done as Research Associate, Post-Doctoral Fellow, Mechanical Engineering Department, The Johns Hopkins University.     

Model of a turbulent boundary layer with explicit identification of the coherent generation structure

Using the method of moments in the space of wavenumbers, a class of models of a developed turbulent flow of an incompressible fluid in a flatplate boundary layer is proposed. The models are based on an analysis of the Navier–Stokes equations that describe the behavior of dynamic coherent structures associated with vorticity generation and also the behavior of the stochastic component. A continuum analog of dynamic equations for a coherent structure is given in an explicit form. In the general case, the stochastic component should satisfy a system of equations of the kinetic type, which reduces to one equation under certain assumptions. It is also shown that the presence of coherent structures leads to generalization of the notion of statistical homogeneity.     

Flow field decomposition applied to experimental data obtained for a transitional boundary layer

Using experimental data from Particle Image Velocimetry (PIV) measurements, coherent structures of a transitional spatially developing boundary layer are determined. The coherent structures are determined utilizing the Proper Orthogonal Decomposition (POD), which is based on an expansion of the flow field variables into a set of eigenfunctions or modes. For having constant and reproducible flow field conditions, the flow is artificially excited by means of periodic velocity fluctuations. The used excitation device allows the generation of different transition scenarios, where this paper focuses on the case of thefundamental transition. Phase locked excitation signals allow the recording of instantaneous velocity fields of the flow field at certain instants of time. It can be shown that PIV is a suitable technique to provide experimental data for POD. The results of the POD show that already a small number of modes cover most of the kinetic energy of the flow.     

Spatiotemporal representation of the dynamic modes in turbulent cavity flows

Proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) were used to extract the coherent structures in turbulent cavity flows. The spatiotemporal representation of the modes was achieved by performing the circular convolution of a change of basis on the data sequence, wherein the transformation function was extracted from the POD or DMD. The spatiotemporal representation of the modes provided significant insight into the evolutionary behavior of the structures. Self-sustained oscillations arise in turbulent cavity flows due to unsteady separation at the leading edge. The turbulent cavity flow at Re_D = 12,000 and a length to depth ratio L/D = 2 was analyzed. The dynamic modes extracted from the data clarified the presence of self-sustained oscillations. The spatiotemporal representation of the POD and DMD modes that caused self-sustained oscillations revealed the prevalent dynamics and evolutionary behavior of the coherent structures from their formation at the leading edge to their impingement at the trailing edge. A local minimum in the mode amplitude representing the energy contributions to the flow was observed upon the impingement of coherent structure at the trailing edge. The modal energy associated with the periodic formation of organized coherent structures followed by their dissipation upon impingement revealed the oscillatory behavior over time.     

剪切湍流大尺度相干结构的模式研究

发展了一种计算剪切湍流大尺度相干结构的新模式．该模式的基础是认为大尺度相干结构为湍流场中流体脉动能量增长最快的那部分，且包含大部分的湍流脉动能量．在此基础上。通过对湍流相干能量方程的推演。建立了描述大尺度相干结构的特征控制方程，并应用Ｃｈｅｂｙｓｈｅｖ多项式方法求得湍流相干能量的最大增长率在波数空间的分布，从而获得对应的大尺度相干结构．应用该模式研究了槽流和一自然对流中的大尺度相干结构，得到的近壁区流动结构与实验现象十分接近．     

The application of classical POD and snapshot POD in a turbulent shear layer with periodic structures

The classical and snapshot proper-orthogonal-decomposition was applied to data taken in a one-stream mixing layer in a narrow channel. Due to this particular geometry the flow develops large periodic structures. POD-analysis of simultaneously measured velocity components in spanwise direction identify as largest mode not only their periodic fraction, but also higher Fourier modes of the two-dimensional fluctuation. The energy content of the plane motion reaches values of about 90%. The amplitude of small three-dimensional vortices embedded in higher POD modes is correlated with the phase of the large structures, which indicates their influence on the entire turbulent motion. Application of scalar snapshot POD on phase averaged data of the entire flow field allows separation into modes. The eigenvalues and eigenvectors show identical distribution for theu- andv-component. Comparison of streakline plots of the reconstructed velocity field from different numbers of modes with flow visualization exhibits that the largest physical structure is described by only the first two modes. This is also supported by calculation of the vorticity component inz-direction. The total energy content of the largest structure is approximately 60%.Supported by the Deutsche Forschungsgemeinschaft under Grant No. Fi 178/28.     

Flow Visualization of Superbursts and of the Log-Layer in a DNS at $$ \operatorname{Re} _{\tau } = 950 $$

This paper uses direct numerical simulations (DNS) of turbulent flow in a channel at (Del álamo, Jiménez, Zandonade, Moser J Fluid Mech 500:135–144, 2004) to provide a picture of the turbulent structures making large contributions to the Reynolds shear stress. Considerable work of this type has been done for the viscous wall region at smaller , for which a log-layer does not exist. Recent PIV measurements of turbulent velocity fluctuations in a plane parallel to the direction of flow have emphasized the dominant contribution of large scale structures in the outer flow. This prompted Hanratty and Papavassiliou (The role of wall vortices in producing turbulence. In: Panton, R.L. (ed) Self-sustaining Mechanism of Wall Turbulence. Computational Mechanics Publications, Southampton, pp. 83–108, 1997) to use DNS at to examine these structures in a plane perpendicular to the direction of flow. They identified plumes which extend from the wall to the center of a channel. The data at are used to explore these results further, to examine the structure of the log-layer, and to test present notions about the viscous wall layer.     

Extended proper orthogonal decomposition: a tool to analyse correlated events in turbulent flows

A tool to analyse correlated events in turbulent flows based on an extended proper orthogonal decomposition (POD) is proposed in this paper. A general definition of extended POD modes is presented and their properties are demonstrated. If the initial POD analysis in a spatio-temporal domain S concerns, for example, velocity—the concept of extended modes can be applied to study the correlation of any physical quantity in any domain with the projection of the velocity field on POD modes in S. The link with particular associations of POD and linear stochastic estimation (LSE) recently proposed is demonstrated at the end of the paper. The method is believed to provide a valuable tool to extend the well-documented POD analysis of eddy structures in turbulent flows, for example, in boundary layers or free shear flows. If extended modes are velocity modes, spatial and temporal interactions between eddy structures can be detected and studied. The rapid development of experimental diagnostic techniques now permit measurements of the concentration in the domain, the velocity of a dispersed phase in the domain or the static pressure at the boundary together with the fluid velocity field. Using this method we are then able to extract objectively the link between the representative groups of velocity modes and the correlated part of the concentration, particle motion or pressure signals.     

Investigation of the spatio-temporal flow structure in the buffer region of a turbulent boundary layer by means of multiplane stereo PIV

In order to examine the physical process associated with the production and transport of turbulence in wall bounded flows, a fully developed turbulent boundary layer flow along a flat plate is investigated in stream-wise span-wise planes at y⁺10, 20, and 30, and Re7,800 by using a high-resolution multiplane stereo PIV system. Of particular interest are the structural features of the coherent flow structures, such as their average size and shape, but their intensity, dynamics, and interaction are also examined. The information is deduced from the joint probability density function of the velocity fluctuations and from various correlation, cross-correlation, and conditional-correlation functions. Furthermore, characteristic instantaneous velocity fields are analyzed in order to examine the importance of the individual coherent flow structures for the production of turbulence and the transport of Reynolds stresses.     

关于湍流拟序结构的思考

本文在简要地回顾了30多年来湍流拟序结构研究的进展之后,评述了目前拟序结构研究的动态。笔者强调准确的湍流场数据库建立的重要性,它是定量描述和研究拟序结构的基础。笔者评述了关于拟序结构目前流行的几种观点和方法,认为应当把研究拟序结构的动力学作为我们研究的目标。笔者深信准确掌握拟序结构的动力学规律,对于预测和控制湍流将有极大的推动。      

A one-dimensional theory of wave-propagation in elastic rods based on the „method of internal constraints“

Conclusions The results obtained in this paper in the particular case of lateral vibrations of bars show an encouraging agreement between the values ofc/c ₀ andc _g/c₀ given by the approximate theory based on the assumption of internal constraints and the exact theory derived from the use of the equations of the Mathematical Theory of Elasticity.This approximate theory which will be referred to as the Theory of Internal Constraints is in the dynamic case completely contained in the constraint equation (1) and in the application ofHamiltons Principle. Accordingly the concept of Shear Coefficient is not used. In the general case of wave propagation in elastic straight rods this theory unifies a number of separate engineering treatments of the problem. Moreover, the same theory can be applied to the study of vibrations of curved bars taking into account the effects of shear and of rotatory inertia as has been shown in a previous paper.The mathematical simplicity of the theory and its degree of accuracy justify its use in dealing with engineering problems in vibrations of curved or straight bars for which more exact theories cannot be used because of their mathematical complexities.The author wants to express his best thanks to Mr.E. C. Zachmanoglou, student in the Department of Aeronautical Engineering, and to Mr.R. V. Milligan, graduate student in the Department of Mechanics, at Rensselaer Polytechnic Institute for the valuable help and assistance given to him in the preparation of the numerical tables and graphs which are presented in this paper.The material presented in this paper is based on an investigation which is being conducted under the sponsorship of the Office of Naval Research, Department of the U. S. Navy, Washington, D. C.; and is presented with the permission of the Office of Naval Research.     

Asymptotic theory of separation flows

A review is given of the results of theoretical investigations of the separation of laminar and turbulent boundary layers in an incompressible fluid obtained on the basis of matched asymptotic expansions valid at large Reynolds numbers (Re). The global picture of the separation flow behind a body of finite size as Re is investigated.Paper presented at Fifth All-Union Symposium on Theoretical and Applied Mechanics, Alma-Ata, 1981.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 20–30, March–April, 1982.     

The dynamics of coherent structures in a flat-plate boundary layer

From the data of a direct numerical simulation the three-dimensional coherent structures of an incompressible, spatially evolving flat-plate boundary layer have been calculated using the POD method. By Galerkin projection of the Navier-Stokes equations onto the corresponding system of eigenfunctions then a low-dimensional model of the flow in the form of a system of ODE's has been derived. In a region of the boundary layer just beyond the spike stages of transition this system displays deterministic chaos that has been quantified by determining its Lyapunov exponents.     

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.      

Streamwise Vortices and Velocity Streaks in a Locally Drag-Reduced Turbulent Boundary Layer

This work aims to understand the changes associated with the near-wall streaky structures in a turbulent boundary layer (TBL) where the local skin-friction drag is substantially reduced. The Reynolds number is R e _?? = 1000 based on the momentum thickness or R e _τ = 440 based on the friction velocity of the uncontrolled flow. The TBL is perturbed via a local surface oscillation produced by an array of spanwise-aligned piezo-ceramic (PZT) actuators and measurements are made in two orthogonal planes using particle image velocimetry (PIV). Data analyses are conducted using the vortex detection, streaky structure identification, spatial correlation and proper orthogonal decomposition (POD) techniques. It is found that the streaky structures are greatly modified in the near-wall region. Firstly, the near-wall streamwise vortices are increased in number and swirling strength but decreased in size, and are associated with greatly altered velocity correlations. Secondly, the velocity streaks grow in number and strength but contract in width and spacing, exhibiting a regular spatial arrangement. Other aspects of the streaky structures are also characterized; they include the spanwise gradient of the longitudinal fluctuating velocity and both streamwise and spanwise integral length scales. The POD analysis indicates that the turbulent kinetic energy of the streaky structures is reduced. When possible, our results are compared with those obtained by other control techniques such as a spanwise-wall oscillation, a spanwise oscillatory Lorentz force and a transverse traveling wave.     

Stability Properties of POD–Galerkin Approximations for the Compressible Navier–Stokes Equations

Fluid flows are very often governed by the dynamics of a mall number of coherent structures, i.e., fluid features which keep their individuality during the evolution of the flow. The purpose of this paper is to study a low order simulation of the Navier–Stokes equations on the basis of the evolution of such coherent structures. One way to extract some basis functions which can be interpreted as coherent structures from flow simulations is by Proper Orthogonal Decomposition (POD). Then, by means of a Galerkin projection, it is possible to find the system of ODEs which approximates the problem in the finite-dimensional space spanned by the POD basis functions. It is found that low order modeling of relatively complex flow simulations, such as laminar vortex shedding from an airfoil at incidence and turbulent vortex shedding from a square cylinder, provides good qualitative results compared with reference computations. In this respect, it is shown that the accuracy of numerical schemes based on simple Galerkin projection is insufficient and numerical stabilization is needed. To conclude, we approach the issue of the optimal selection of the norm, namely the H ¹ norm, used in POD for the compressible Navier–Stokes equations by several numerical tests. Received 21 April 1999 and accepted 18 November 1999