Modelling of Marangoni convection using proper orthogonal decomposition

Proper orthogonal decomposition (POD) is applied to Marangoni convection in a horizontal fluid layer heated from below and cooled from above with non-deformable free surface. We investigate two-dimensional Marangoni convection for the case of free-slip bottom in the limit of small Prandtl number. The POD technique is then used to the velocity and temperature data to obtain basis functions for both velocity and temperature fields. When these basis functions are used in a Galerkin procedure, the low-dimensional of Marangoni convection are constructed with the smallest possible degree of freedom. The results based on this low-dimensional model are discussed.     

The performance of proper orthogonal decomposition in discontinuous flows

In this paper,flow reconstruction accuracy and flow prediction capability of discontinuous transonic flow field by means of proper orthogonal decomposition(POD) method is studied.Although linear superposition of ‘‘high frequency waves' ' in different POD modes can achieve the reconstruction of the shock wave,the smoothness of the solution near the shock wave cannot be guaranteed.The modal coefficients are interpolated or extrapolated and different modal components are superposed to realize the prediction of the flow field beyond the snapshot sets.Results show that compared with the subsonic flow,the transonic flow with shock wave requires more POD modes to reach a comparative reconstruction accuracy.When a shock wave exists,the interpolation prediction ability is acceptable.However,large errors exist in extrapolation,and increasing the number of POD modes cannot effectively improve the prediction accuracy of the flow field.     

On the hidden beauty of the proper orthogonal decomposition

The proper orthogonal decomposition theorem (Loève, 1955) of probability theory has been proposed by Lumley (1967, 1972, 1981) for detection of spatial coherent patterns in turbulent flows. More specifically, the decomposition extracts deterministic functions from second-order statistics of a random field and converges optimally fast in quadratic mean (i.e., in energy). The technique can be made completely deterministic in the sense that it can be applied to spatially and temporally evolving flows. The remarkable property of this deterministic decomposition is not only in its optimal convergence (as emphasized before) but also in its space/time symmetry which permits access to the spatiotemporal dynamics. The flow is decomposed into both spatial and temporal orthogonal modes which are coupled: each space component is associated with a time component partner. The latter is the time evolution of the former and the former is the spatial configuration of the latter. This generalizes the notion of spatial and temporal structures which can be followed through the various instabilities that the flow undergoes as Reynolds number increases. It also provides a nonlinear dynamics tool for spatiotemporal dynamical systems and can be used for bifurcation detection and analysis as well as dimension and degree of complexity estimates.Dedicated to Professor J.L. Lumley on the occasion of his 60th birthday.This work was supported by an NSF/PYI award MSS89-57462, and partially by a NATO Grant No. 900265 which are gratefully acknowledged.     

基于本征正交分解的网格加筋筒壳模型降阶方法

针对网格加筋筒壳结构动力响应分析效率低的问题,论文提出了一种基于本征正交分解技术的模型降阶方法.基本思路是通过静力分析获得原模型的节点位移场并组装成快照矩阵,利用本征正交分解技术提取快照矩阵的主成分作为转换矩阵,实现模型降阶.通过算例对比验证了论文提出的降阶模型具有较高的计算精度及效率,降阶模型的低阶频率计算结果与全阶模型十分吻合,高阶频率误差仅为1.01%,而计算时间为全阶模型的0.03%.最后以自由-固支的网格加筋筒为例,采用降阶模型计算其在不同激励下的振动响应,降阶模型的计算结果与全阶模型非常吻合,计算效率有明显提升.     

The use of dynamic basis functions in proper orthogonal decomposition

Currently, the usefulness of proper orthogonal decomposition (POD) is limited to computational domains with fixed meshes and fixed boundaries. This paper presents a new POD method that enables the modeling of flow through computational domains with deforming meshes and/or moving boundaries. To achieve this goal, the solution is approximated using basis functions which, although not explicitly functions of time, depend on parameters associated with flow unsteadiness. Results are shown for transonic flow through the Tenth Standard Configuration. Comparisons are made between this method and the standard approach for on- and off-reference flow conditions. This method properly captured flow nonlinearities and shock motion for cases in which the classical POD method failed.     

A proper orthogonal decomposition of a simulated supersonic shear layer

The Karhunen–Loève procedure is applied to the analysis of an ensemble of snapshots obtained from a conditionally sampled localized shear layer simulation. The computed set of optimal basis functions is used to economically characterize sampled flow realizations. Pictorially it is seen that the essential features (and roughly 80% of the energy) of typical flows are captured by retaining roughly 10–20 parameters in the expansion. Smaller-scale features are resolved by retaining more terms in the series.     

A procedure based on proper orthogonal decomposition for time-frequency analysis of time series

A procedure for time-frequency analysis of time series is described, which is mainly inspired by singular-spectrum analysis, but it presents some modifications that allow checking the convergence of the results and extracting the detected spectral components through a more efficient technique, especially for real applications. This technique is adaptive, completely data dependent with no a priori assumption and applicable to non-stationary signals. The principal components are extracted from the signals and sorted by their fluctuating energy; moreover, the time variation of their amplitude and frequency is characterized. The technique is first assessed for multi-component computer-generated signals and then applied to experimental velocity signals. The latter are acquired in proximity of the wake generated from a triangular prism placed vertically on a plane, with a vertical edge against the incoming flow. From these experimental signals, three different spectral components, connected to the dynamics of different vorticity structures, are detected, and the time histories of their amplitudes and frequencies are characterized.     

Reduced-order proper orthogonal decomposition extrapolating finite volume element format for two-dimensional hyperbolic equations

This paper is concerned with establishing a reduced-order extrapolating finite volume element(FVE) format based on proper orthogonal decomposition(POD) for two-dimensional(2D) hyperbolic equations. For this purpose, a semi discrete variational format relative time and a fully discrete FVE format for the 2D hyperbolic equations are built, and a set of snapshots from the very few FVE solutions are extracted on the first very short time interval. Then, the POD basis from the snapshots is formulated,and the reduced-order POD extrapolating FVE format containing very few degrees of freedom but holding sufficiently high accuracy is built. Next, the error estimates of the reduced-order solutions and the algorithm procedure for solving the reduced-order format are furnished. Finally, a numerical example is shown to confirm the correctness of theoretical conclusions. This means that the format is efficient and feasible to solve the2 D hyperbolic equations.     

Low-order modelling of shallow water equations for sensitivity analysis using proper orthogonal decomposition

This article presents a reduced-order model (ROM) of the shallow water equations (SWEs) for use in sensitivity analyses and Monte-Carlo type applications. Since, in the real world, some of the physical parameters and initial conditions embedded in free-surface flow problems are difficult to calibrate accurately in practice, the results from numerical hydraulic models are almost always corrupted with uncertainties. The main objective of this work is to derive a ROM that ensures appreciable accuracy and a considerable acceleration in the calculations so that it can be used as a surrogate model for stochastic and sensitivity analyses in real free-surface flow problems. The ROM is derived using the proper orthogonal decomposition (POD) method coupled with Galerkin projections of the SWEs, which are discretised through a finite-volume method. The main difficulty of deriving an efficient ROM is the treatment of the nonlinearities involved in SWEs. Suitable approximations that provide rapid online computations of the nonlinear terms are proposed. The proposed ROM is applied to the simulation of hypothetical flood flows in the Bordeaux breakwater, a portion of the ‘Rivière des Prairies' located near Laval (a suburb of Montreal, Quebec). A series of sensitivity analyses are performed by varying the Manning roughness coefficient and the inflow discharge. The results are satisfactorily compared to those obtained by the full-order finite volume model.     

Visualization of coherent structure in scalar fields of unsteady jet flows with interferometric tomography and proper orthogonal decomposition

Proper orthogonal decomposition (POD) is a method of examining spatial coherence in unsteady flow fields from an ensemble of multidimensional measurements. When applied to experimental data, the proper orthogonal decomposition is generally restricted to data sets with low spatial resolution. This is because of the inherent difficulties in generating an ensemble of measurements that contain a large number of data points. In this paper, a system for obtaining a large ensemble of three-dimensional scalar measurements using interferometric tomography is presented. The proper orthogonal decomposition is applied in three spatial dimensions to experimental data of two jet-like flows. The coherent structure present in the near field of a neutrally buoyant, helium–argon jet and the far field of a buoyant helium jet into air is visualized. The POD results of the helium–argon jet clearly reveal the breakdown region of a sequence of vortex rings and a large-scale flapping motion in the jet far field. The POD of the buoyant helium jet shows a number of competing modes with varying degrees of helicity. Received: 14 January 2000/Accepted: 26 September 2000     

A variable‐fidelity aerodynamic model using proper orthogonal decomposition

A variable‐fidelity aerodynamic model based on proper orthogonal decomposition (POD) of an ensemble of computational fluid dynamics (CFD) solutions at different parameters is presented in this article. The ensemble of CFD solutions consists of two subsets of numerical solutions or snapshots computed at two different nominal orders of accuracy or discretization. These two subsets are referred to as the low‐fidelity and high‐fidelity solutions or data, whereby the low fidelity corresponds with computations made at the lower nominal order of accuracy or coarser discretization. In this model, the relatively inexpensive low‐fidelity data and the more accurate but expensive high‐fidelity data are considered altogether to devise an efficient prediction methodology involving as few high‐fidelity analyses as possible, while obtaining the desired level of detail and accuracy. The POD of this set of variable‐fidelity data produces an optimal linear set of orthogonal basis vectors that best describe the ensemble of numerical solutions altogether. These solutions are projected onto this set of basis vectors to provide a finite set of scalar coefficients that represent either the low‐fidelity or high‐fidelity solutions. Subsequently, a global response surface is constructed through this set of projection coefficients for each basis vector, which allows predictions to be made at parameter combinations not in the original set of observations. This approach is used to predict supersonic flow over a slender configuration using Navier–Stokes solutions that are computed at two different levels of nominal accuracy as the low‐fidelity and high‐fidelity solutions. The numerical examples show that the proposed model is efficient and sufficiently accurate. Copyright © 2016 John Wiley & Sons, Ltd.     

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.     

Analysis of misfires in a direct injection engine using proper orthogonal decomposition

Previous work demonstrated that the occasional misfired and partially burned cycles (MF) in a stratified-charge, spark-ignited direct injection engine always achieved an early flame kernel, but failed to reach and inflame the fuel in the bottom of the piston bowl. This conclusion was derived from intra-cycle crank angle resolved velocity and fuel concentration images that were recorded simultaneously using high-speed particle image velocimetry and planar laser-induced fluorescence. In this study, both ensemble average analysis, conditionally sampled on either MF or Well Burned (WB) cycles and proper orthogonal decomposition (POD) are applied separately to the velocity and fuel distributions. POD of the velocity and fuel distributions near the spark plug were performed, and the mode energy and structure of the modes are compared. This analysis is used to assess the similarity and differences between the MF and the WB cycles and to identify physical insight gained by POD. The POD modes were determined from the combined set of 200 WB and 37 MF cycles to create two sets of 237 orthogonal modes, one set for the velocity, V, and one for the equivalence ratio, ε. Then, conditionally sampled averages of the POD coefficients could be used to quantify the extent to which each mode contributed to the MFs. Also, the probability density functions of the coefficients quantified the cyclic variability of each mode’s contribution. The application of proper orthogonal decomposition to velocity and equivalence ratio images was useful in identifying and analyzing the differences in flow and mixture conditions at the time of spark between well-burning and misfiring cycles. However, POD results alone were not sufficient to identify which of the cycles were misfiring cycles, and additional information was required for conditional sampling.     

Reduced order model of offshore wind turbine wake by proper orthogonal decomposition

A wide range of previously designed methods for faster parametrization of partial differential equations requires them to be solved using existing finite volume, finite element, and finite difference solvers. Due to the requirement of high degrees of freedom to accurately model the physical system, computational costs often becomes a bottle-neck. It poses challenges to conducting efficient repeated parametric sampling of the input parameter that disrupts the whole design process. Model reduction techniques adopted to high fidelity systems provide a basis to accurately represent a physical system with a lower degree of freedom. The present work focuses on one such method for high-fidelity simulations that combines finite volume strategy with proper orthogonal decomposition and Galerkin projection to test reduced-order models for high Reynolds number flow applications. The model is first benchmarked against flow around a cylinder for which extensive numerical and experimental data is available in the literature. The models are then tested to full-scale NREL 5MW offshore wind turbines to evaluate wake evolution in the downstream direction. The simulations results show relative errors of wind turbines for the first seventy modes approach 4.7% in L²-norm for velocities.     

Model reduction for supersonic cavity flow using proper orthogonal decomposition (POD) and Galerkin projection

The reduced-order model(ROM) for the two-dimensional supersonic cavity flow based on proper orthogonal decomposition(POD) and Galerkin projection is investigated. Presently, popular ROMs in cavity flows are based on an isentropic assumption,valid only for flows at low or moderate Mach numbers. A new ROM is constructed involving primitive variables of the fully compressible Navier-Stokes(N-S) equations, which is suitable for flows at high Mach numbers. Compared with the direct numerical simulation(DNS) results, the proposed model predicts flow dynamics(e.g., dominant frequency and amplitude) accurately for supersonic cavity flows, and is robust. The comparison between the present transient flow fields and those of the DNS shows that the proposed ROM can capture self-sustained oscillations of a shear layer. In addition, the present model reduction method can be easily extended to other supersonic flows.