风力机翼型气动计算的降阶模型研究

将基于POD的降阶模型应用于风力机翼型的气动研究。首先应用CFD数值模拟得到一系列快照结果;应用基于本征正交分解(POD)的方法得到流场的一组基模态,认为对于所研究的问题,任一流场可以由这些基模态通过线性表达得到;对控制方程进行Galerkin投影,得到降阶模型,将离散求解N-S方程的问题转化为一组只有十几个自由度的常微分方程,从而减少计算时间,提高计算效率。并对二维翼型的绕流的定常和非定常问题进行了分析,计算结果表明,降阶模型可以较好地捕捉流动的特征,与直接CFD模拟相比计算精度相当,但大幅有效地提高了计算效率。     

Reliable reduced-order models for time-dependent linearized Euler equations

Development of optimal reduced-order models for linearized Euler equations is investigated. Recent methods based on proper orthogonal decomposition (POD), applicable for high-order systems, are presented and compared. Particular attention is paid to the link between the choice of the projection and the efficiency of the reduced model. A stabilizing projection is introduced to induce a stable reduced-order model at finite time even if the energy of the physical model is growing. The proposed method is particularly well adapted for time-dependent hyperbolic systems and intrinsically skew-symmetric models. This paper also provides a common methodology to reliably reduce very large nonsymmetric physical problems.     

Projection-free proper orthogonal decomposition method for a cantilever plate in supersonic flow

This paper explores the use of the proper orthogonal decomposition (POD) method for supersonic nonlinear flutter of a cantilever plate or wing. The aeroelastic equations are constructed using von Karman plate theory and first-order piston theory. The two-dimensional POD modes (POMs) in xy plane are determined from the chaotic results given by the traditional Rayleigh–Ritz (RR) approach. For a specific structure, the POMs need to be calculated once and then can be used for various parameters of interest. The derivatives of the POMs are calculated numerically to avoid the complex projection from the POMs to the Rayleigh–Ritz modes (RRMs). Numerical examples demonstrate that the POD method using 4 POMs can obtain accurate limit cycle oscillation (LCO) results with substantial computational cost savings, compared with 12 RRMs by the Rayleigh–Ritz method. The POD method is employed for the analysis of the chaotic oscillations. It is also demonstrated that the POD modes are robust over a range of flight parameters.     

近壁湍流的降阶模型及应用

由条带和流向涡的循环再生构成的近壁自维持过程(self-sustaining process, SSP)是壁湍流产生和维持的重要机制. 文章通过对最小槽道的直接数值模拟(direct numerical simulation, DNS)获得近壁自维持过程的流场数据, 采用正规正交分解法(proper orthogonal decomposition, POD)对该数据进行分析, 获得了不同流向和展向尺度的特征模态, 通过将Navier-Stokes方程在这些模态上进行投影, 得到近壁自维持过程的降阶模型, 并采用DNS数据对降阶模型的预测能力进行了评价. 该模型被初步应用于大涡模拟近壁模型的构造.      

Two-level discretizations of nonlinear closure models for proper orthogonal decomposition

Proper orthogonal decomposition has been successfully used in the reduced-order modeling of complex systems. Its original promise of computationally efficient, yet accurate approximation of coherent structures in high Reynolds number turbulent flows, however, still remains to be fulfilled. To balance the low computational cost required by reduced-order modeling and the complexity of the targeted flows, appropriate closure modeling strategies need to be employed. Since modern closure models for turbulent flows are generally nonlinear, their efficient numerical discretization within a proper orthogonal decomposition framework is challenging. This paper proposes a two-level method for an efficient and accurate numerical discretization of general nonlinear closure models for proper orthogonal decomposition reduced-order models. The two-level method computes the nonlinear terms of the reduced-order model on a coarse mesh. Compared with a brute force computational approach in which the nonlinear terms are evaluated on the fine mesh at each time step, the two-level method attains the same level of accuracy while dramatically reducing the computational cost. We numerically illustrate these improvements in the two-level method by using it in three settings: the one-dimensional Burgers equation with a small diffusion parameter ν = 10^?3, the two-dimensional flow past a cylinder at Reynolds number Re = 200, and the three-dimensional flow past a cylinder at Reynolds number Re = 1000.     

An improved algorithm for balanced POD through an analytic treatment of impulse response tails

We present a modification of the balanced proper orthogonal decomposition (balanced POD) algorithm for systems with simple impulse response tails. In this new method, we use dynamic mode decomposition (DMD) to estimate the slowly decaying eigenvectors that dominate the long-time behavior of the direct and adjoint impulse responses. This is done using a new, low-memory variant of the DMD algorithm, appropriate for large datasets. We then formulate analytic expressions for the contribution of these eigenvectors to the controllability and observability Gramians. These contributions can be accounted for in the balanced POD algorithm by simply appending the impulse response snapshot matrices (direct and adjoint, respectively) with particular linear combinations of the slow eigenvectors. Aside from these additions to the snapshot matrices, the algorithm remains unchanged. By treating the tails analytically, we eliminate the need to run long impulse response simulations, lowering storage requirements and speeding up ensuing computations. To demonstrate its effectiveness, we apply this method to two examples: the linearized, complex Ginzburg–Landau equation, and the two-dimensional fluid flow past a cylinder. As expected, reduced-order models computed using an analytic tail match or exceed the accuracy of those computed using the standard balanced POD procedure, at a fraction of the cost.     

Bilinear modal representations for reduced-order modeling of localized piecewise-linear oscillators

A novel reduced-order modeling method for vibration problems of elastic structures with localized piecewise-linearity is proposed. The focus is placed upon solving nonlinear forced response problems of elastic media with contact nonlinearity, such as cracked structures and delaminated plates. The modeling framework is based on observations of the proper orthogonal modes computed from nonlinear forced responses and their approximation by a truncated set of linear normal modes with special boundary conditions. First, it is shown that a set of proper orthogonal modes can form a good basis for constructing a reduced-order model that can well capture the nonlinear normal modes. Next, it is shown that the subspace spanned by the set of dominant proper orthogonal modes can be well approximated by a slightly larger set of linear normal modes with special boundary conditions. These linear modes are referred to as bi-linear modes, and are selected by an elaborate methodology which utilizes certain similarities between the bi-linear modes and approximations for the dominant proper orthogonal modes. These approximations are obtained using interpolated proper orthogonal modes of smaller dimensional models. The proposed method is compared with traditional reduced-order modeling methods such as component mode synthesis, and its advantages are discussed. Forced response analyses of cracked structures and delaminated plates are provided for demonstrating the accuracy and efficiency of the proposed methodology.     

Reduced-order models for parameter dependent geometries based on shape sensitivity analysis

The proper orthogonal decomposition (POD) is widely used to derive low-dimensional models of large and complex systems. One of the main drawback of this method, however, is that it is based on reference data. When they are obtained for one single set of parameter values, the resulting model can reproduce the reference dynamics very accurately but generally lack of robustness away from the reference state. It is therefore crucial to enlarge the validity range of these models beyond the parameter values for which they were derived. This paper presents two strategies based on shape sensitivity analysis to partially address this limitation of the POD for parameters that define the geometry of the problem at hand (design or shape parameters.) We first detail the methodology to compute both the POD modes and their Lagrangian sensitivities with respect to shape parameters. From them, we derive improved reduced-order bases to approximate a class of solutions over a range of parameter values. Secondly, we demonstrate the efficiency and limitations of these approaches on two typical flow problems: (1) the one-dimensional Burgers’ equation; (2) the two-dimensional flows past a square cylinder over a range of incidence angles.     

System identification-guided basis selection for reduced-order nonlinear response analysis

Reduced-order nonlinear simulation is often times the only computationally efficient means of calculating the extended time response of large and complex structures under severe dynamic loading. This is because the structure may respond in a geometrically nonlinear manner, making the computational expense of direct numerical integration in physical degrees of freedom prohibitive. As for any type of modal reduction scheme, the quality of the reduced-order solution is dictated by the modal basis selection. The techniques for modal basis selection currently employed for nonlinear simulation are ad hoc and are strongly influenced by the analyst's subjective judgment. This work develops a reliable and rigorous procedure through which an efficient modal basis can be chosen. The method employs proper orthogonal decomposition to identify nonlinear system dynamics, and the modal assurance criterion to relate proper orthogonal modes to the normal modes that are eventually used as the basis functions. The method is successfully applied to the analysis of a planar beam and a shallow arch over a wide range of nonlinear dynamic response regimes. The error associated with the reduced-order simulation is quantified and related to the computational cost.     

基于POD模态分解的液环泵瞬态气液两相流分析

针对液环泵内气液两相流动的复杂时空规律,采用本征正交分解(POD)方法对其瞬态气液两相流场进行特征分解,分析其相态场、速度场的空间基模态特征及模态系数的时域特征,建立非定常流场降阶模型,并对流场进行预测分析。结果表明POD方法可实现对液环泵内复杂流场的时空解耦分析,相态场及速度场的各阶模态系数在时域内的变化能够反映各阶模态场的能量、频率及相位变化规律,模态场能够反映脉动流场的空间尺度变化规律。POD降阶模型能够对样本空间内的流场进行精确预测,进口压力、相态场及速度场预测结果的最大相对误差分别约为0.2%、4%、8%,在样本空间外POD降阶模型具有一定外延预测精度,当预测目标逐渐远离样本空间时POD预测结果与CFD计算结果之间的误差逐渐增大。     

Visualisation and analysis of large-scale vortex structures in three-dimensional turbulent lid-driven cavity flow

In this paper, large eddy simulation (LES) of a three-dimensional turbulent lid-driven cavity (LDC) flow at Re = 10,000 has been performed using the multiple relaxation time lattice Boltzmann method. A Smagorinsky eddy viscosity model was used to represent the sub-grid scale stresses with appropriate wall damping. The prediction for the flow field was first validated by comparing the velocity profiles with previous experimental and LES studies, and then subsequently used to investigate the large-scale three-dimensional vortical structures in the LDC flow. The instantaneous three-dimensional coherent structures inside the cavity were visualised using the second invariant (Q), Δ criterion, λ₂ criterion, swirling strength (λ_ci) and streamwise vorticity. The vortex structures obtained using the different criteria in general agree well with each other. However, a cleaner visualisation of the large vortex structures was achieved with the λ_ci criterion and also when the visualisation is based on the vortex identification criteria expressed in terms of the swirling strength parameters. A major objective of the study was to perform a three-dimensional proper orthogonal decomposition (POD) on the fluctuating velocity fields. The higher energy POD modes efficiently extracted the large-scale vortical structures within the flow which were then visualised with the swirling strength criterion. Reconstruction of the instantaneous fluctuating velocity field using a finite number of POD modes indicated that the large-scale vortex structures did effectively approximate the large-scale motion. However, such a reduced order reconstruction of the flow based on the large-scale vortical structures was clearly not as effective in predicting the small-scale details of the fluctuating velocity field which relate to the turbulent transport.     

Modal reduction of mathematical models of biological molecules

This paper reports a detailed study of modal reduction based on either linear normal mode (LNM) analysis or proper orthogonal decomposition (POD) for modeling a single α-d-glucopyranose monomer as well as a chain of monomers attached to a moving atomic force microscope (AFM) under harmonic excitations. Also a modal reduction method combining POD and component modal synthesis is developed. The accuracy and efficiency of these methods are reported. The focus of this study is to determine to what extent these methods can reduce the time and cost of molecular modeling and simultaneously provide the required accuracy. It has been demonstrated that a linear reduced order model is valid for small amplitude excitation and low frequency excitation. It is found that a nonlinear reduced order model based on POD modes provides a good approximation even for large excitation while the nonlinear reduced order model using linear eigenmodes as the basis vectors is less effective for modeling molecules with a strong nonlinearity. The reduced order model based on component modal synthesis using POD modes for each component also gives a good approximation. With the reduction in the dimension of the system using these methods the computational time and cost can be reduced significantly.     

抛物化Navier-Stokes方程的降维仿真模型

利用特征投影分解技术和奇值分解方法,建立抛物化Navier-Stokes方程的降维仿真模型,讨论降维仿真模型解的稳定性和误差,并利用误差估计指导特征投影分解基函数的选取及特征投影分解基函数的更新.最后,用数值试验验证降维仿真模型的有效性和可行性.     

Enhanced linearized reduced-order models for subsurface flow simulation

Trajectory piecewise linearization (TPWL) represents a promising approach for constructing reduced-order models. Using TPWL, new solutions are represented in terms of expansions around previously simulated (and saved) solutions. High degrees of efficiency are achieved when the representation is projected into a low-dimensional space using a basis constructed by proper orthogonal decomposition of snapshots generated in a training run. In recent work, a TPWL procedure applicable for two-phase subsurface flow problems was presented. The method was shown to perform well for many cases, such as those with no density differences between phases, though accuracy and robustness were found to degrade in other cases. In this work, these limitations are shown to be related to model accuracy at key locations and model stability. Enhancements addressing both of these issues are introduced. A new TPWL procedure, referred to as local resolution TPWL, enables key grid blocks (such as those containing injection or production wells) to be represented at full resolution; i.e., these blocks are not projected into the low-dimensional space. This leads to high accuracy at selected locations, and will be shown to improve the accuracy of important simulation quantities such as injection and production rates. Next, two techniques for enhancing the stability of the TPWL model are presented. The first approach involves a basis optimization procedure in which the number of columns in the basis matrix is determined to minimize the spectral radius of an appropriately defined amplification matrix. The second procedure incorporates a basis matrix constructed using snapshots from a simulation with equal phase densities. Both approaches are compatible with the local resolution procedure. Results for a series of test cases demonstrate the accuracy and stability provided by the new treatments. Finally, the TPWL model is used as a surrogate in a direct search optimization algorithm, and comparison with results using the full-order model demonstrates the efficacy of the enhanced TPWL procedures for this application.     

POD-based wall boundary conditions for the numerical simulation of turbulent channel flows

Simulation of turbulent wall-bounded flows requires a high spatial resolution in the wall region, which limits the range of Reynolds numbers which can be effectively reached. In previous work, we proposed proper orthogonal decomposition (POD) based wall boundary conditions to bypass the simulation of the inner wall region. Tests were carried out for direct numerical simulation at a low Reynolds number Re_τ = 180. The boundary condition is based on the POD spatial eigenfunctions which are determined a priori in the full channel. It consists of a three-component velocity field on the plane y⁺ = 50 which is reconstructed at each instant from a combination of selected eigenfunctions. The coefficients of the combination are estimated from the simulation in the reduced domain using the threshold-based reconstruction method described in Podvin et al. The study is now extended to large-eddy simulation at higher Reynolds numbers Re_τ = 295 and Re_τ = 590. Two versions of the reconstruction method are considered. In the first version, both the phases and the moduli of the coefficients are allowed to vary. In the second version, only the phases are adjusted. We find that the latter method is associated with improved statistics and is relatively robust with respect to the reconstruction threshold. However, it is sensitive to the details of the numerical simulation, unlike the former method, which is associated with less accurate statistics and is more dependent on the reconstruction threshold.     

飞行器大攻角复杂流动的POD和DMD对比分析

基于非结构/混合网格、耗散自适应2阶混合格式以及脱体涡模拟（detached eddy simulation,DES）方法开展了现代战斗机模型复杂分离流动的数值模拟,并与有限的平均气动力试验数据进行了对比,结果表明计算具有合理性,在此基础上进一步应用本征正交分解（proper orthogonal decomposition,POD）和动力学模态分解（dynamic mode decomposition,DMD）方法对数值模拟流场的非定常特性进行了对比分析.研究表明飞行器背风区流场由一对边条涡的螺旋运动主导,旋涡破裂前在横向空间截面上流场是中性稳定的,同时主涡核的运动是多频耦合的.POD和DMD的对比分析则表明:两者模态配对的方式不同,但主要模态之间具有一定相关性;POD模态中包含多种频率的运动,而且能量较集中于主模态,流场重构效率更高;DMD则将流场的主要特征运动提取为一些单频模态的组合,同时能够给出模态的稳定性.      

基于POD方法的高超声速边界层转捩判定方法

高超声速边界层转捩是高超声速飞行器设计的关键基础问题之一.为了研究高超声速边界层转捩, 在风洞中, 对平板模型进行了M=5的实验, 在模型中心沿流动方向使用PCB脉动压力传感器对脉动压力时间序列进行采集.文章将本征正交分解(proper orthogonal decomposition,POD)方法引入高超声速脉动压力数据处理中, 发展了单点POD分析方法.经验证, 使用该方法重构数据的均方根(root mean square, RMS)峰值位置可表征转捩位置, 实用性强.      

Predictive flow-field estimation

We are considering the problem of real-time prediction of 3D turbulent velocity fields based on a small number of scalar measurements. The method of proper orthogonal decomposition (POD) allows for the decomposition of an ensemble of velocity fields into a set of spatial basis functions and a set of temporal coefficients. The computation of the temporal coefficients is by no means a trivial matter, especially when one is faced with a large number of modes. In this paper we discuss the use of radial basis function (RBF) models to capture the discrete time evolution and nonlinear dynamics of the POD coefficients. Further, we propose the use of regularized regression techniques to generate models that provide mappings between the POD coefficients and scalar measurements. As a final step towards real-time prediction, the state-space RBF models and regression measurement models are combined using unscented Kalman filters to produce optimal solutions such that a balance between the state models and measurement models is achieved.The proposed methods are tested for two specific cases. The classical Lorenz model is chosen to demonstrate the use and effectiveness of RBF models as a potential candidate for state models. Flow around a wall-mounted cube in a channel at Re=20,000 is considered as the second case. The aim for the second case is to be able to accurately predict the POD coefficients outside the ensemble. It is shown that a large number of POD coefficients is required to approximate the velocity fields with sufficient accuracy. The RBF models are created based on only the temporal information available from the initial ensemble, and it is shown that the RBF model is able to correctly approximate the high-dimensional phase space. Combined with the unscented Kalman filter it is indeed possible to track the evolution of the POD coefficients for a long time. The robustness of the filter is demonstrated by considering the presence of noise in measurements and using measurement information at time steps greater than the evolution time step.