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.     

An improved algorithm for the shallow water equations model reduction: Dynamic Mode Decomposition vs POD

We propose an improved framework for dynamic mode decomposition (DMD) of 2‐D flows for problems originating from meteorology when a large time step acts like a filter in obtaining the significant Koopman modes, therefore, the classic DMD method is not effective. This study is motivated by the need to further clarify the connection between Koopman modes and proper orthogonal decomposition (POD) dynamic modes. We apply DMD and POD to derive reduced order models (ROM) of the shallow water equations. Key innovations for the DMD‐based ROM introduced in this paper are the use of the Moore–Penrose pseudoinverse in the DMD computation that produced an accurate result and a novel selection method for the DMD modes and associated amplitudes and Ritz values. A quantitative comparison of the spatial modes computed from the two decompositions is performed, and a rigorous error analysis for the ROM models obtained is presented. Copyright © 2015 John Wiley & Sons, Ltd.     

POD analysis of a finite-length cylinder near wake

The near wake of a wall-mounted finite-length square cylinder with an aspect ratio of 7 is investigated based on the proper orthogonal decomposition (POD) of the PIV data measured in three spanwise planes, i.e., z/d = 6, 3.5 and 1.0, near the cylinder free end, mid-span and fixed end (wall), respectively. The Reynolds number based on free-stream velocity (U _∞) and cylinder width (d) is 9,300. A two-dimensional (2D) square cylinder wake is also measured and analyzed at the same Reynolds number for the purpose of comparison. The structures of various POD modes show marked differences between the two flows. While the coefficients, a ₁ and a ₂, of the POD modes 1 and 2 occur within an annular area centered at a ₁ = a ₂ = 0 in the 2D wake, their counterparts are scattered all over the entire circular plane at z/d = 1.0 and 3.5 of the finite-length cylinder wake. Flow at z/d = 6 is dominated by POD mode 1, which corresponds to symmetrical vortex shedding and accounts for 54.0 % of the total turbulent kinetic energy (TKE). On the other hand, the POD modes 1 and 2, corresponding to anti-symmetrical vortex shedding, are predominant, accounting for about 45.0 % of the total TKE, at z/d = 3.5 and 1. It has been found that the flow structure may be qualitatively and quantitatively characterized by the POD coefficients. For example, at z/d = 6, a larger a ₁ corresponds to a smaller length of flow reversal zone and a stronger downwash flow. At z/d = 3.5 and 1, two typical flow modes can be identified from a ₁ and a ₂. While large a ₁ and/or a ₂ correspond to anti-symmetrical vortex shedding, as in a 2D cylinder wake, small a ₁ and a ₂ lead to symmetrical vortex shedding. Any values between the large and small a ₁ and/or a ₂ correspond to the flow structure between these two typical flow modes. As such, the probability of occurrence of a flow structure may be determined from the distribution of the POD coefficients.     

An indirect shooting method based on the POD/DEIM technique for distributed optimal control of the wave equation

This paper presents a fast numerical method, based on the indirect shooting method and Proper Orthogonal Decomposition (POD) technique, for solving distributed optimal control of the wave equation. To solve this problem, we consider the first‐order optimality conditions and then by using finite element spatial discretization and shooting strategy, the solution of the optimality conditions is reduced to the solution of a series of initial value problems (IVPs). Generally, these IVPs are high‐order and thus their solution is time‐consuming. To overcome this drawback, we present a POD indirect shooting method, which uses the POD technique to approximate IVPs with smaller ones and faster run times. Moreover, in the presence of the nonlinear term, to reduce the order of the nonlinear calculations, a discrete empirical interpolation method (DEIM) is applied and a POD/DEIM indirect shooting method is developed. We investigate the performance and accuracy of the proposed methods by means of 4 numerical experiments. We show that the presented POD and POD/DEIM indirect shooting methods dramatically reduce the CPU time compared to the full indirect shooting method, whereas there is no significant difference between the accuracy of the reduced and full indirect shooting methods.     

POD enriched boundary models and their optimal stabilisation

Modes obtained using the Proper Orthogonal Decomposition are used as boundary enrichment functions within a variational multiscale method for the stochastically forced Burgers equation. Initially, large increases in accuracy are obtained using the enrichment functions without stabilisation terms. Then, optimal coefficients for the stabilisation parameter τ of the unresolved scale model are calibrated using a goal‐oriented model‐constrained optimisation technique, resulting in further improvements. As both the determination of the enrichment functions and the optimisation of the coefficients requires high‐accuracy reference data, a scaling procedure is introduced to allow their use over range of conditions. Numerical experiments confirm that the scaling procedure is effective. Copyright © 2014 John Wiley & Sons, Ltd.     

On investigation of particle dispersion by a POD approach

The aim of this communication is to show the ability of POD to compute the instantaneous flow velocity when applying the Lagrangian technique to predict particle dispersion. The instantaneous flow velocity at the particle's location is obtained by solving a low-order dynamical model, deduced by a Galerkin projection of the Navier-Stokes equations onto each POD eigenfunction and it is coupled with the particle's equation of motion. This technique is applied to particle dispersion in a three-dimensional lid driven cavity. It yields a substantial decrease in computing time in comparison with LES computation and it enables treating different cases of particle dispersion Published in Prikladnaya Mekhanika, Vol. 44, No. 1, pp. 133–142, January 2008.     

Gappy POD方法重构湍流数据的研究

Gappy POD 是一种基于本征正交分解(proper orthogonal decomposition, POD)的数据重构方法. 本文研究了gappy POD在湍流数据重构中的应用, 主要关注了以下两个因素的影响: 第一, 数据本身的复杂程度, 即构成流场的POD模态数量; 第二, 破损区域的面积大小和几何形状. 考虑到上述因素, 本文重新严格地表述了gappy POD的重构过程, 并推导出gappy POD重构误差的公式. 论文选取旋转湍流数据为案例进行了gappy POD重构的研究, 并解释了构成gappy POD重构误差的两个部分. 第一部分来自流场POD展开的截断误差, 该截断误差会被POD基函数在已知点上的值组成的矩阵的最小特征值放大. 这部分误差主要取决于流场的复杂程度, 当流场复杂程度较低时, 相应误差随采用的POD模态数目增大而减小. 当流场复杂程度较高时, 很小的POD截断误差也会导致很大的重构误差, 此时需要采用流场所有的POD模态进行重构以消除截断误差. 重构误差的第二部分来自POD基函数在已知点上的值组成的矩阵的非列满秩性, 它主要取决于破损区域的面积大小和几何形状. 破损区域的面积越大, 或者破损面积相同时, 破损区域内信息所包含的相关性越大, 第二部分的重构误差越大.      

利用POD对双坡屋盖风压场预测

本文利用本征正交分解(POD)方法对双坡屋盖未布置测压点位置的风压时间序列进行了预测。采用线性插值及双线性插值得到预测点位置上的本征模态值。结构由原风压场协方差分析得到的主坐标和上述新本征模态值获得未布置测压点位置的风压时间序列。通过在时间域和频率域内比较预测出的风压与实测风压说明了POD在预测双坡屋盖风压场的有效性。     

The near wake of sinusoidal wavy cylinders: Three-dimensional POD analyses

Three-dimensional (3D) proper orthogonal decomposition (POD) analyses are conducted to investigate the near wake of sinusoidal wavy cylinders. For a wave amplitude a/D_m = 0.152, three typical spanwise wavelengths (λ_z) of the wavy cylinder are taken into account, i.e., λ_z/D_m = 1.89, 3.79 and 6.06, where D_m is the mean diameter of the wavy cylinder, among which λ_z/D_m = 1.89 and 6.06 are the optimum wavelengths corresponding to the largest reduction/suppression of fluid forces acting on the wavy cylinder. Time- and space-resolved three-component velocities of the near wake flow, obtained from large eddy simulation (LES) at a subcritical Reynolds number Re = 3 × 10³, are used in the 3D POD analyses. Comparison is made among the wavy cylinders of the three λ_z/D_m values as well as between them and a smooth cylinder, in terms of POD modes, mode energy, mode coefficients, as well as reconstructed flow structures by lower modes. For the optimum λ_z/D_m = 1.89 and 6.06, energy associated with the first two POD modes is significantly reduced compared with that for λ_z/D_m = 3.79 and the smooth cylinder. Distinct characteristics are observed on the lower POD modes for the wavy cylinders. It is found that the first two POD modes for λ_z/D_m = 1.89 and 6.06 are linked to large-scale streamwise vortices that are additionally introduced into the near wake due to the wavy geometry. Meanwhile, POD mode 3 suggests that the wavy cylinder with the larger optimum λ_z/D_m (= 6.06) generates dominant hairpin-like and spanwise coherent structures (CSs) shedding from the saddle at a different frequency from those shedding from the node. Evolutionary development of these CSs is discussed based on reconstructed flows.     

Analysis of different POD methods for PIV-measurements in complex unsteady flows

Proper Orthogonal Decomposition (POD) is an effective tool in fluid dynamics for investigation of complex, transitional or turbulent flows. In POD the transient vector or scalar field (velocity, concentration, temperature, etc.) is decomposed into a sum of spatial modes multiplied with time coefficients (Fourier-splitting method). However, these spatial modes and time coefficients can in practice be obtained by different methods. Even if POD has been used in numerous fluid dynamical studies, there are only few publications describing the relationship between the different methods and comparing the results. In the present case the POD basis functions are calculated either by Singular Value Decomposition (SVD) or by the Snapshot-POD approach. The results are compared in order to understand similarities and differences between the methods, as well as advantages and drawbacks. Comparisons between the obtained spatial modes, time coefficients, required computational effort, and complexity of calculation are presented and discussed. The influence of the numerical settings is also investigated, in particular the impact of the number of snapshots on the results. Finally, the differences obtained when analyzing a vector field globally or component-wise are discussed in detail.     

An application of pattern mapping to plane motion

A new, highly automated method for measuring plane motion, pattern mapping, has been developed for rigidbody motion and strain analysis. Pattern mapping employs image processing and syntactic pattern recognition principles to recognize a known pattern before and after motion. In this paper, the Lagrangian definition of motion was used to map points in the two images, and the map was used to determine plane motion. Full-image analysis is thoroughly demonstrated.     

An application of Ungar's differential transform to elastodynamics

In recent years, a lot of writers have used Cagniard-de Hoop’s method^[1][2] to solve some problems of elastic wave. But it is a difficult and complicated task to change the path of integration when we use this method. A differential transform by A.Ungar^[3,6] can obviate this difficulty. In this paper, weuse Ungar’s differential transform to solve a case of Lamb’s problem^[1][2].     

A study of inner-outer interactions in turbulent channel flows by interactive POD

The amplitude and frequency modulation of near-wall flow structures by the large-scale motions in outer regions is studied in turbulent channel flows. The proper orthogonal decomposition(POD) method is applied to investigate the interactions between the near-wall motions and the large-scale flow modes of the outer regions based on two datasets from direct numerical simulation of turbulent channel flows at Reynolds numbers of 550–10 0 0. The fluctuations in the fields u~+, v~+, w~+ and Reynolds shear stress-(uv)~+ are studied to understand the mechanism of amplitude and frequency modulation of the nearwall structures by the outer large-scale motions. The amplitude modulation coefficient of the Reynolds shear stress is larger than that of the velocity components. The frequency modulation effect has an opposite influence in the spanwise direction compared to the streamwise direction. The streamwise characteristic frequency increases with increasing large-scale velocity. However, the spanwise characteristic frequency exhibits a decreasing trend with increasing large-scale velocity in the near-wall region.     

Bifurcation analysis of reduced rotor model based on nonlinear transient POD method

The singularity theory is applied to study the bifurcation behaviors of a reduced rotor model obtained by nonlinear transient POD method in this paper. A six degrees of freedom (DOFs) rotor model with cubically nonlinear stiffness supporting at both ends is established by the Newton's second law. The nonlinear transient POD method is used to reduce a six-DOFs model to a one-DOF one. The reduced model reserves the dynamical characteristics and occupies most POM energy of the original one. The singularity of the reduced system is analyzed, which replaces the original system. The bifurcation equation of the reduced model indicates that it is a high co-dimension bifurcation problem with co-dimension 6, and the universal unfolding (UN) is provided. The transient sets of six unfolding parameters, the bifurcation diagrams between the bifurcation parameter and the state variable are plotted. The results obtained in this paper present a new kind of method to study the UN theory of multi-DOFs rotor system.     

Filtered density function modelling of near-wall scalar transport with POD velocity modes

This paper addresses the problem of heat transfer in the near-wall region of a turbulent channel flow. We investigate the role of coherent eddy structures on the thermal exchange between the solid and fluid materials. In the present approach, the instantaneous velocity field is modelled by means of a low-order dynamical system based on proper orthogonal decomposition (POD) modes while the temperature field is accounted for by using a stochastic filtered density function (FDF) method. A good comparison of results with reference DNS data on mean temperature and on temperature r.m.s. at various Prandtl numbers and for different wall boundary conditions (isotemperature, isoflux, and conjugate heat transfer) is achieved.     

An application of nonsymmetric Lax-Milgram lemma to nonassociated plasticity

Based on a general assumption for plastic potential and yield surface, some properties of the nonassociated plasticity are studied, and the existence and uniqueness of the distribution of incremental stress and displacement for work-hardening materials are proved by using nonsymmetric Lax-Milgram lemma, when the work-hardening parameter A>F/Q/–F/, Q/.     

POD analysis of flow structures in a scale model of a ventilated room

Measurements with particle image velocimetry have been carried out in a scale model of the Annex 20 room. Data were taken in a plane near the inlet. The flow consisted of a wall jet (Re=5,000) and a low-velocity region below the jet. POD was used to analyze dominant flow structures. The analysis showed that the flow some of the time has flow structures very different from the mean velocity field. A time-resolved data series was projected onto the orthonormal basis derived from the POD for analysis of the time variation of the POD amplitudes.