Dynamic mode decomposition and its applications in fluid dynamics
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摘要: 随着计算流体力学和先进流动测试技术的发展，流动的刻画越来越精细，伴随而来的海量流场信息的模态提取与复杂动力学特征的模型化成为当前流体力学的研究热点。动力学模态分解（Dynamic Mode Decomposition，DMD）作为一个全新的时空耦合型动力学建模方法，得到迅速推广。DMD是一种数据驱动的非定常流场模态分析手段，可以准确捕捉各个流动模态的频率及增长特性，并建立流场演化的动力学降阶模型，以重构或预测流场动力学过程。本文针对DMD在流体力学研究的应用问题，重点综述了DMD算法自提出以来的一系列改进以及对不同流动现象的应用，并通过典型测试算例说明DMD的应用过程。在此基础上，讨论了DMD的研究现状及未来发展方向。Abstract: With the development of computational fluid dynamics, the revelation for the flow structure in unsteady flows becomes much increasingly delicate. This brings a mass of flow information and catalyzes the study of mode extraction to analyze complex dynamic behaviors. This review discusses a representative approach for flow mode extraction, called dynamic mode decomposition (DMD). DMD is a novel technique for modeling flow dynamics from both spatial and temporal data, which becomes popular recently. As a data-driven algorithm, DMD is capable of capturing the frequency and growth rate of flow modes, helping to construct efficient reduced-order models for flow analysis and control. The availability of DMD has been shown in many complex flow phenomena, like turbulence and transition. To improve its robustness, different methodologies have been introduced, including sparsity-promoting, compressive sensing, time-delayed embedding, etc. Moreover, DMD shows a close relationship with Koopman theory (describing the dynamics of a nonlinear system by an infinite-dimensional linear operator) and proper orthogonal decomposition (a well-known technique for analyzing fluid data). In the present paper, the efficacy of DMD has been shown by two test cases:1) identification of a low-dimensional system, 2) analysis of transonic buffet phenomenon. Furthemore, the future development of DMD is discussed.
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Key words:
- dynamic mode decomposition /
- reduced-order model /
- unsteady flow /
- Koopman operator /
- data-driven
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图 1 跨声速NACA0012翼型升力系数随时间的响应
Figure 1. Lift coefficient response of a NACA0012 airfoil in transonic flow
图 2 某时刻压力云图及观测点
Figure 2. Pressure contour and observation points at an instantaneous time
图 7 采样段195.8时刻真实流场与重构流场对比
Figure 7. Real and reconstructed flow fields at sampling time 195.8
图 8 预测段232.4时刻真实流场与重构流场对比
Figure 8. Real and reconstructed flow fields at predicting time 232.4
图 10 流场中观测点C、D、E压强随时间变化的预测结果
Figure 10. Temporal evolution of local pressures at observation points C, D and E
表 1 改进的DMD算法总结
Table 1. Overview of improved DMD algorithms
改进类型 具体操作 算法改进 残差最小化：优化DMD(Optimized DMD, opt-DMD)[12]，通过全局优化算法，迭代计算出最佳的特征值及对应的主模态；最优模态分解(Optimal Mode Decomposition, OMD)[21]，将(15)转化为带子空间正交性约束的最小化问题；稀疏增强DMD(Sparsity-promoting DMD, SPDMD)[22]，最小化问题(16)分为模态选择和振幅修正两步求解。 系统矩阵计算：流动DMD(Streaming DMD)[23]，随着样本增加而在线更新系统矩阵，降低标准DMD存储量；总体DMD (Total DMD, TDMD)[24]，利用总体最小二乘方法消除偏差，获得准确的系统矩阵。 改进模态特性：递归DMD(Recursive DMD, RDMD)[25]，平衡POD模态的正交性和低截断误差，以及DMD的单频特征；谱POD(Spectral POD, SPOD)[26]，发展一种兼顾低能量和多频率特征的模态分解方法，关联低残差和纯频率的模态特征[27]。 模态系数修正：参数化DMD(Parametrized DMD)[28]，假设标准DMD的模态系数存在误差，在SPDMD基础上对幅值进行时间积分；非嵌入式线性模型[29]，用径向基函数插值，对选择的DMD模态系数演化进行预测。 频率捕捉：多分辨率DMD(Multiresolution DMD, mrDMD)[30]，对样本逐层进行DMD，增强频率捕捉能力；高阶DMD(High Order DMD, HODMD)[31]，处理有限维空间内的多频率动力学系统。 流动预测过程：标准DMD的预测模型与Kalman滤波过程相结合，实现非定常流场计算过程中的在线预测[32]；DMD和Galerkin投影相结合的非定常流场降阶模型[33]。 采样环境影响：噪声修正的DMD算法[34]；不满足Nyquist-Shannon采样定理的数据分析[35]；适应非均匀采样的DMD算法(Non-Uniform DMD, NU-DMD)[36]；压缩感知与DMD算法结合[37]。 非线性拓展：扩展DMD(Extending DMD, EDMD)[38]，对快照样本做非线性预处理；基于核方法的DMD[39]，缓解EDMD的高维问题。 模态选择 主要模态选择：快照序列投影到辨识的动态模态上得到各个模态的幅值[40]；与模态范数和特征值相关的能量准则[41]；模态范数与其频率的逆的加权作为各个模态的能量[42]；模态系数在采样区域内的积分[43]；矢量过滤准则用于模态选择[44]。 其他应用 CFD数值模拟过程加速收敛[45]。辨识外输入系统：带控制的DMD(Dynamic Mode Decomposition with Control, DMDc) [46]；输入输出DMD(Input-Output Dynamic Mode Decomposition, IODMD)[47]；带控制系统的EDMD[48]；基于Koopman理论的带外输入建模方法[49]。 
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表 2 DMD在流体力学问题中的应用
Table 2. Applications of DMD in fluid dynamics
类型 具体应用及特点 台阶流动 粒子图像测速(Particle Image Velocimetry, PIV)试验环境下的低速后台阶流动，DMD与OMD对比[21]
PIV试验环境下，方形管道的后台阶流动，POD与DMD对比[65]
改进的延迟脱体涡模拟(Improved Delayed Detached Eddy Simulation, IDDES)仿真的后台阶流动[66]
PIV试验环境下，涡流发生器的后台阶流动，DMD用于POD重构的流场[67]
带状雷诺平均Navier-Stokes/大涡模拟(Zonal Reynolds Averaged Navier-Stokes-Large Eddy Simulation, Zonal RANS-LES)的太空发射器后台阶流动[68-70]
LES数值模拟的平板前台阶流动，POD与DMD对比[71]方腔流动 直接数值模拟(Direct Numerical Simulation, DNS)数值模拟的雷诺数4500方腔流动[9]
LES数值模拟的方腔自激振荡，POD与DMD对比[72]
二维试验和三维数值模拟的方腔流动，DMD在高维和任意采样系统问题的应用[36]
多尺度有限元方法数值模拟的多孔媒介中的流体，POD和DMD对比[73]
DNS数值模拟的L型方腔流动，DMD与线性稳定性分析方法对比[74]
风洞试验和混合RANS/LES数值模拟的矩形方腔流动，POD与DMD对比[75]
DNS数值模拟的开放式方腔饱和过程[76]
DNS数值模拟下，有控制和无控制的高超声速方腔流[77]
不同大小的高超声速开放式方腔流，POD与DMD对比[78]
超燃冲压发动机凹腔流动，POD与DMD对比[79]射流 PIV试验环境下，双圆柱间射流[9]
DNS数值模拟的横向射流[10]
PIV试验的雷诺数5000横向喷水射流[40]
LES数值模拟的高超声速管射流及PIV试验环境下，双圆柱间射流，SPDMD[22]
数值模拟的燃烧射流；PIV试验环境下，层流反对称射流[80]
PIV试验环境下的氦气射流[19]
PIV试验环境下，方腔内的湍流合成射流，POD与DMD对比[81]
RANS数值模拟的横向射流[82]
LES数值模拟的斜向冷却射流[83]
LES数值模拟的厚壁孔洞射流[84, 85]
LES数值模拟的空间发展横向射流，POD与DMD对比[86]
LES数值模拟下，横流中的高速射流[87]
DNS数值模拟下，低速横向射流的剪切层特性[88]
阴影法试验环境下，有无声场的横向剪切射流[89]燃烧 基于线化NS方程数值模拟的带挡板三维燃烧室[90]
PIV试验环境下，低速和高速旋转的火焰喷射，POD与DMD对比[91]
LES数值模拟的燃气轮机燃烧不稳定性[92]
LES数值模拟的低速旋转火焰中，流动火焰耦合问题[93]
试验环境下，多孔燃烧室和钝头体后反应流，POD与DMD对比[94]
LES数值模拟的矩形燃烧室中液体燃料燃烧[95]
LES数值模拟的涡轮机叶片间接燃烧噪声，DMD与SPDMD的应用[96]
试验研究中，火焰燃烧的分叉过程，DMD与Parametrized DMD对比[28]
PIV试验环境下，预混倾斜燃烧室的间歇性振荡[97]
PIV试验环境下，湍流旋转反应流动的动态结构[98]简单对称外形 PIV试验的柔性膜绕流尾迹分析[9]
浸入边界法数值模拟的椭圆前缘有限厚度平板，带控制和不带控制情况，POD与DMD对比[99]
PIV试验的椭圆前缘有限厚度平板[15]
LES数值模拟的大迎角旋成体低频涡脱流动[100]
LES数值模拟的悬臂梁绕流尾迹，POD和DMD对比[20]
浸入边界法数值模拟的柔性旗帜流-固-热耦合现象[101]
LES数值模拟的开环控制钝头体尾迹[102]
DNS数值模拟和PIV试验的大迎角半球绕流圆筒，POD与DMD对比[103, 104]
PIV试验的噪声环境下的椭圆前缘有限厚度平板分离控制，TDMD分析[24]/TDMD和Streaming DMD结合[105]
PIV试验的有限长度钝头板分离流动[106]
PIV试验的平板层流分离现象，POD和DMD对比[107]柱/球体绕流 浸入边界法数值模拟的雷诺数60圆柱绕流，DMD与Opt-DMD对比[12]
DNS数值模拟的尾迹区带控制的圆柱绕流[15]
PIV试验的雷诺数413圆柱绕流，Streaming DMD[23]，不满足Nyquist采样定理的DMD修正[35]，基于核方法的Koopman谱分析[39]
DNS数值模拟的亚临界圆柱涡致振动特征分析[108]
DNS数值模拟的脉冲横向射流控制下的圆柱[109]
DNS数值模拟的雷诺数50的圆柱绕流，Koopman模态与DMD算法的关联[110]
PIV试验的边界层附近圆柱尾迹流动[111]
PIV试验的单圆柱和两个不同大小并列圆柱的尾迹，POD与DMD对比[112]
PIV试验的雷诺数13000圆柱绕流尾迹分析，DMD与优化振幅的DMD对比[41]
谱元法数值模拟的低雷诺数椭圆柱尾迹[113]
浸入边界法数值模拟的串列双圆柱近壁面效应，前圆柱带控制[114]
PIV试验的合成射流圆柱绕流控制，Fourier模态分解、POD与DMD对比[115]
浸入边界法数值模拟的圆柱绕流[116]
PIV试验的雷诺数1000旋转圆柱绕流[117]
DNS数值模拟的二维圆柱和三维球体绕流，全局稳定性分析、POD与DMD对比[118]
DNS数值模拟的三维无限展长圆柱绕流，DMD与Galerkin投影结合[119]
DNS数值模拟的单静止圆柱和三旋转圆柱绕流，POD，Opt-DMD和RDMD对比[25]
PIV试验的波浪形圆柱尾迹[120]
PIV试验的圆柱绕流，带合成射流控制，基于TDMD[121]
数值模拟和试验数据的圆柱绕流，几种噪声修正DMD方法对比[34]
DNS数值模拟的三维球体绕流降阶模型，DMD与Galerkin投影结合[33]
DNS数值模拟的雷诺数60圆柱绕流，DMD与改进振幅选择的DMD对比[43]
DNS数值模拟的绕方柱纳米流体[122]湍流与转捩 DNS数值模拟的二维平板Poiseuille流动，SPDMD[22]
PIV试验的平板发卡涡生成，POD与DMD对比[123]
LES数值模拟的激波-湍流边界层干扰[124]
DNS数值模拟的黏弹性流体湍流转捩[125]
LES数值模拟的风力机翼尖涡不稳定，POD与DMD对比[126]
DNS数值模拟的转捩后期近壁面边界层[127]
DNS数值模拟的高速转捩问题[128]
PIV试验的湍流涡环演化，应用POD、DMD与SPDMD[129]
PIV试验的内燃机缸内湍流场[130]
DNS数值模拟的低频非定常激波-湍流边界层干扰[131]
隐式LES数值模拟的高超声速边界层转捩[132]
LES数值模拟的激波-湍流边界层干扰，全局线性稳定性分析与DMD对比[133]
PIV实验的不稳定驻点流动[134]
基于eN方法的空间DMD转捩预测[135]
非定常扩压通道内有控和无控分离流[136]机翼流动 试验和数值模拟环境下的动失速分析，POD与DMD对比[137]
LES数值模拟的沉浮自由度翼型动失速，POD与DMD对比[138]
LES数值模拟的沉浮自由度翼型动失速，DMD与经验模态分解(Empirical Mode Decomposition, EMD)对比[139]
PIV试验的带Gurney襟翼NACA0015翼型尾流[140-141]
DNS数值模拟的SD7003翼段转捩尾流分析[142]
浸入边界法数值模拟的大迎角翼型周期俯仰尾迹分析[143]
风洞试验的NACA0012翼型任意运动气动力建模，采用DMDc方法[144]
RANS数值模拟的跨声速抖振，极限环段POD与DMD对比[145]；过渡段DMD、SPDMD与改进振幅选择的DMD对比[43]其他流动现象 LES数值模拟的吹吸气流动控制[146]
基于数值模拟的热对流循环问题[147]
DES数值模拟的高速火车流动结构，POD与DMD对比[148]
LES数值模拟的风力机尾迹建模，DMD与Kalman滤波结合[32]
LES数值模拟的带外输入风力机流场建模，IODMD[47]
RANS数值模拟的无叶扩压器非定常流动[149]
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表 3 主要DMD模态的增长率和频率
Table 3. Growth rates and frequencies of dominant DMD modes
DMD mode Growth rate Reduced frequency 1 0 0 2 0.0156 0.1973 3 0.0259 0.2192 4 0.0478 0 5 0.0434 0.4005
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[1] 张来平, 邓小刚, 何磊, 等. E级计算给CFD带来的机遇与挑战[J].空气动力学学报, 2016, 34(4):405-417. http://www.kqdlxxb.com/CN/abstract/abstract11872.shtmlZhang L P, Deng X G, He L, et al. The opportunity and grand challenges in computational fluid dynamics by exascale computing[J]. Acta Aerodynamica Sinica, 2016, 34(4):405-417. (in Chinese) http://www.kqdlxxb.com/CN/abstract/abstract11872.shtml [2] 张涵信.关于CFD高精度保真的数值模拟研究[J].空气动力学学报, 2016, 34(1):1-4. http://www.kqdlxxb.com/CN/abstract/abstract11764.shtmlZhang H X. Investigations on fidelity of high order accurate numerical simulation for computational fluid dynamics[J]. Acta Aerodynamica Sinica, 2016, 34(1):1-4. (in Chinese) http://www.kqdlxxb.com/CN/abstract/abstract11764.shtml [3] Ghoreyshi M, Jirásek A, Cummings R M. Reduced order unsteady aerodynamic modeling for stability and control analysis using computational fluid dynamics[J]. Progress in Aerospace Sciences, 2014, 71:167-217. doi: 10.1016/j.paerosci.2014.09.001 [4] Dowell E H, Hall K C. Modeling of fluid-structure interaction[J]. Annual Review of Fluid Mechanics, 2001, 33(1):445-490. doi: 10.1146/annurev.fluid.33.1.445 [5] Lucia D J, Beran P S, Silva W A. Reduced-order modeling:new approaches for computational physics[J]. Progress in Aerospace Sciences, 2004, 40(1-2):51-117. doi: 10.1016/j.paerosci.2003.12.001 [6] 张伟伟, 叶正寅.基于CFD的气动力建模及其在气动弹性中的应用[J].力学进展, 2008, 38(1):77-86. doi: 10.6052/1000-0992-2008-1-J2006-158Zhang W W, Ye Z Y. On unsteady aerodynamic modeling based on CFD technique and its applications on aeroelastic analysis[J]. Advances in Mechanics, 2008, 38(1):77-86. (in Chinese) doi: 10.6052/1000-0992-2008-1-J2006-158 [7] Brunton S L, Noack B R. Closed-loop turbulence control:progress and challenges[J]. Applied Mechanics Reviews, 2015, 67(5):050801. doi: 10.1115/1.4031175 [8] Schmid P J, Sesterhenn J. Dynamic mode decomposition of numerical and experimental data[C]//Sixty-First Annual Meeting of the APS Division of Fluid Dynamics, San Antonio, Texas, USA, 2008: 208. [9] Schmid P J. Dynamic mode decomposition of numerical and experimental data[J]. Journal of Fluid Mechanics, 2010, 656:5-28. doi: 10.1017/S0022112010001217 [10] Rowley C W, Mezić I, Bagheri S, et al. Spectral analysis of nonlinear flows[J]. Journal of Fluid Mechanics, 2009, 641:115-127. doi: 10.1017/S0022112009992059 [11] Mezić I. Spectral properties of dynamical systems, model reduction and decompositions[J]. Nonlinear Dynamics, 2005, 41(1):309-325. doi: 10.1007/s11071-005-2824-x.pdf [12] Chen K K, Tu J H, Rowley C W. Variants of dynamic mode decomposition:boundary condition, Koopman, and Fourier analyses[J]. Journal of Nonlinear Science, 2012, 22(6):887-915. doi: 10.1007/s00332-012-9130-9 [13] Mezić I. Analysis of fluid flows via spectral properties of the Koopman operator[J]. Annual Review of Fluid Mechanics, 2013, 45(1):357-378. doi: 10.1146/annurev-fluid-011212-140652 [14] Rowley C W, Dawson S T M. Model reduction for flow analysis and control[J]. Annual Review of Fluid Mechanics, 2017, 49:387-417. doi: 10.1146/annurev-fluid-010816-060042 [15] Tu J H, Rowley C W, Luchtenburg D M, et al. On dynamic mode decomposition:theory and applications[J]. Journal of Computational Dynamics, 2014, 1(2):391-421. doi: 10.3934/jcd [16] Duke D, Soria J, Honnery D. An error analysis of the dynamic mode decomposition[J]. Experiments in Fluids, 2012, 52(2):529-542. doi: 10.1007/s00348-011-1235-7 [17] Bagheri S. Effects of weak noise on oscillating flows:linking quality factor, Floquet modes, and Koopman spectrum[J]. Physics of Fluids, 2014, 26(9):094104. doi: 10.1063/1.4895898 [18] Pan C, Xue D, Wang J J. On the accuracy of dynamic mode decomposition in estimating instability of wave packet[J]. Experiments in Fluids, 2015, 56(8):164. doi: 10.1007/s00348-015-2015-6 [19] Schmid P J, Li L, Juniper M P, et al. Applications of the dynamic mode decomposition[J]. Theoretical and Computational Fluid Dynamics, 2011, 25(1-4):249-259. doi: 10.1007/s00162-010-0203-9 [20] Cesur A, Carlsson C, Feymark A, et al. Analysis of the wake dynamics of stiff and flexible cantilever beams using POD and DMD[J]. Computers & Fluids, 2014, 101:27-41. http://cn.bing.com/academic/profile?id=ecabfe5b3636e0a1551e6dd2d22ab468&encoded=0&v=paper_preview&mkt=zh-cn [21] Wynn A, Pearson D S, Ganapathisubramani B, et al. Optimal mode decomposition for unsteady flows[J]. Journal of Fluid Mechanics, 2013, 733:473-503. doi: 10.1017/jfm.2013.426 [22] Jovanović M R, Schmid P J, Nichols J W. Sparsity-promoting dynamic mode decomposition[J]. Physics of Fluids, 2014, 26(2):024103. doi: 10.1063/1.4863670 [23] Hemati M S, Williams M O, Rowley C W. Dynamic mode decomposition for large and streaming datasets[J]. Physics of Fluids, 2014, 26(11):111701. doi: 10.1063/1.4901016 [24] Hemati M S, Rowley C W, Deem E A, et al. De-biasing the dynamic mode decomposition for applied Koopman spectral analysis of noisy datasets[J]. Theoretical and Computational Fluid Dynamics, 2017, 31(4):349-368. doi: 10.1007/s00162-017-0432-2 [25] Noack B R, Stankiewicz W, Morzyński M, et al. Recursive dynamic mode decomposition of transient and post-transient wake flows[J]. Journal of Fluid Mechanics, 2016, 809:843-872. doi: 10.1017/jfm.2016.678 [26] Sieber M, Paschereit C O, Oberleithner K. Spectral proper orthogonal decomposition[J]. Journal of Fluid Mechanics, 2016, 792:798-828. doi: 10.1017/jfm.2016.103 [27] Noack B R. From snapshots to modal expansions-bridging low residuals and pure frequencies[J]. Journal of Fluid Mechanics, 2016, 802:1-4. doi: 10.1017/jfm.2016.416 [28] Sayadi T, Schmid P J, Richecoeur F, et al. Parametrized data-driven decomposition for bifurcation analysis, with application to thermo-acoustically unstable systems[J]. Physics of Fluids, 2015, 27(3):037102. doi: 10.1063/1.4913868 [29] Bistrian D A, Navon I M. Randomized dynamic mode decomposition for non-intrusive reduced order modelling[J]. International Journal for Numerical Methods in Engineering, 2017, 112(1):3-25. doi: 10.1002/nme.v112.1 [30] Kutz J N, Fu X, Brunton S L. Multiresolution dynamic mode decomposition[J]. SIAM Journal on Applied Dynamical Systems, 2016, 15(2):713-735. doi: 10.1137/15M1023543 [31] Le Clainche S, Vega J M. High order dynamic mode decomposition[J]. SIAM Journal on Applied Dynamical Systems, 2017, 16(2):882-925. doi: 10.1137/15M1054924 [32] Iungo G V, Santoni-Ortiz C, Abkar M, et al. Data-driven reduced order model for prediction of wind turbine wakes[J]. Journal of Physics:Conference Series, 2015, 625(1):83-90. http://cn.bing.com/academic/profile?id=44ed3b8c3f498473bc20cfb083871d4c&encoded=0&v=paper_preview&mkt=zh-cn [33] Stankiewicz W, Morzyński M, Kotecki K, et al. On the need of mode interpolation for data-driven Galerkin models of a transient flow around a sphere[J]. Theoretical and Computational Fluid Dynamics, 2017, 31(2):111-126. doi: 10.1007/s00162-016-0408-7 [34] Dawson S T M, Hemati M S, Williams M O, et al. Characterizing and correcting for the effect of sensor noise in the dynamic mode decomposition[J]. Experiments in Fluids, 2016, 57(3):42-61. doi: 10.1007/s00348-016-2127-7 [35] Tu J H, Rowley C W, Kutz J N, et al. Spectral analysis of fluid flows using sub-Nyquist-rate PIV data[J]. Experiments in Fluids, 2014, 55(9):1805. doi: 10.1007/s00348-014-1805-6 [36] Guéniat F, Mathelin L, Pastur L R. A dynamic mode decomposition approach for large and arbitrarily sampled systems[J]. Physics of Fluids, 2015, 27(2):025113. doi: 10.1063/1.4908073 [37] Brunton S L, Proctor J L, Tu J H, et al. Compressed sensing and dynamic mode decomposition[J]. Journal of Computational Dynamics, 2016, 2(2):165-191. doi: 10.3934/jcd [38] Williams M O, Kevrekidis I G, Rowley C W. A data-driven approximation of the Koopman operator:extending dynamic mode decomposition[J]. Journal of Nonlinear Science, 2015, 25(6):1307-1346. doi: 10.1007/s00332-015-9258-5 [39] Williams M O, Rowley C W, Kevrekidis I G. A kernel-based method for data-driven Koopman spectral analysis[J]. Journal of Computational Dynamics, 2017, 2(2):247-265. https://arxiv.org/pdf/1411.2260v4 [40] Schmid P J, Violato D, Scarano F. Decomposition of time-resolved tomographic PIV[J]. Experiments in Fluids, 2012, 52(6):1567-1579. doi: 10.1007/s00348-012-1266-8 [41] Tissot G, Cordier L, Benard N, et al. Model reduction using dynamic mode decomposition[J]. Comptes Rendus Mécanique, 2014, 342(6-7):410-416. doi: 10.1016/j.crme.2013.12.011 [42] Bistrian D A, Navon I M. An improved algorithm for the shallow water equations model reduction:dynamic mode decomposition vs POD[J]. International Journal for Numerical Methods in Fluids, 2015, 78(9):552-580. doi: 10.1002/fld.v78.9 [43] Kou J Q, Zhang W W. A criterion to select dominant modes of dynamic mode decomposition[J]. European Journal of Mechanics-B/Fluids, 2017, 62:109-129. doi: 10.1016/j.euromechflu.2016.11.015 [44] Bistrian D A, Navon I M. The method of dynamic mode decomposition in shallow water and a swirling flow problem[J]. International Journal for Numerical Methods in Fluids, 2017, 83(1):73-89. doi: 10.1002/fld.v83.1 [45] Andersson N. A non-intrusive acceleration technique for compressible flow solvers based on dynamic mode decomposition[J]. Computers & Fluids, 2016, 133:32-42. https://www.sciencedirect.com/science/article/pii/S0045793016301219 [46] Proctor J L, Brunton S L, Kutz J N. Dynamic mode decomposition with control[J]. SIAM Journal on Applied Dynamical Systems, 2016, 15(1):142-161. doi: 10.1137/15M1013857 [47] Annoni J, Seiler P. A method to construct reduced-order parameter-varying models[J]. International Journal of Robust and Nonlinear Control, 2016, 27(4):582-597. https://www.aem.umn.edu/~SeilerControl/Papers/2017/AnnoniEtAl_17IJRNC_AMethodToConstructReducedOrderLPVModels.pdf [48] Williams M O, Hemati M S, Dawson S T M, et al. Extending data-driven Koopman analysis to actuated systems[C]//Proceedings of the 10th IFAC Symposium on Nonlinear Control Systems. Marriott Hotel Monterey, California, USA, 2016: 704-709. [49] Proctor J L, Brunton S L, Kutz J N. Including inputs and control within equation-free architectures for complex systems[J]. The European Physical Journal Special Topics, 2016, 225(13):2413-2434. doi: 10.1140/epjst/e2016-60057-9.pdf [50] Koopman B O. Hamiltonian systems and transformation in Hilbert space[J]. Proceedings of the National Academy of Sciences, 1931, 17(5):315-318. doi: 10.1073/pnas.17.5.315 [51] Mezić I, Banaszuk A. Comparison of systems with complex behavior[J]. Physica D, 2004, 197(1-2):101-133. doi: 10.1016/j.physd.2004.06.015 [52] Budisic M, Mohr R, Mezic I. Applied Koopmanism[J]. Chaos, 2012, 22(4):047510. doi: 10.1063/1.4772195 [53] 贾继莹. Koopman算符在一些动力系统中的算法和应用研究[D]. 北京: 清华大学, 2015.Jia J Y. Research on the algorithms and applications of Koopman operator on some dynamical systems[D]. Beijing: Tsinghua University, 2015. (in Chinese) [54] Taira K, Brunton S L, Dawson S T M, et al. Modal analysis of fluid flows:an overview[J]. AIAA J, 2017, 55(12):4013-4041. doi: 10.2514/1.J056060 [55] Williams M O, Rowley C W, Mezić I, et al. Data fusion via intrinsic dynamic variables:an application of data-driven Koopman spectral analysis[J]. Europhysics Letters, 2015, 109(4):40007. doi: 10.1209/0295-5075/109/40007 [56] Brunton S L, Brunton B W, Proctor J L, et al. Koopman invariant subspaces and finite linear representations of nonlinear dynamical systems for control[J]. PLoS ONE, 2016, 11(2):e0150171. doi: 10.1371/journal.pone.0150171 [57] Holmes P, Lumley J L, Berkooz G, et al. Turbulence, coherent structures, dynamical systems and symmetry[M]. Cambridge University Press, 2012. [58] Berkooz G, Holmes P, Lumley J L. The proper orthoghnal decomposition in the analysis of turbulent flows[J]. Annual Review of Fluid Mechanics, 1993, 25(1):539-575. doi: 10.1146/annurev.fl.25.010193.002543 [59] 王汉封, 徐胜金.用POD方法研究有限长正方形棱柱尾流的双稳态现象[J].空气动力学学报, 2014, 32(6):827-833. http://www.kqdlxxb.com/CN/abstract/abstract11579.shtmlWang H F, Xu S J. On the bistable phenomenon of a finite length square cylinder wake with POD method. Acta Aerodynamica Sinica, 2014, 32(6):827-833. (in Chinese) http://www.kqdlxxb.com/CN/abstract/abstract11579.shtml [60] 董圣华, 史爱明, 叶正寅, 等.超临界翼型跨声速抖振CFD计算和POD分析[J].空气动力学学报, 2015, 33(4):481-487. http://www.kqdlxxb.com/CN/abstract/abstract11697.shtmlDong S H, Shi A M, Ye Z Y, et al. CFD computation and POD analysis for transonic buffet on a supercritical airfoil[J]. Acta Aerodynamica Sinica, 2015, 33(4):481-487. (in Chinese) http://www.kqdlxxb.com/CN/abstract/abstract11697.shtml [61] Barocio E, Pal B C, Thornhill N F, et al. A dynamic mode decomposition framework for global power system oscillation analysis[J]. IEEE Transactions on Power Systems, 2015, 30(6):2902-2912. doi: 10.1109/TPWRS.2014.2368078 [62] Cui L X, Long W. Trading strategy based on dynamic mode decomposition:tested in Chinese stock market[J]. Physica A:Statistical Mechanics and its Applications, 2016, 461:498-508. doi: 10.1016/j.physa.2016.06.046 [63] Hua J C, Roy S, McCauley J L, et al. Using dynamic mode decomposition to extract cyclic behavior in the stock market[J]. Physica A:Statistical Mechanics and its Applications, 2016, 448:172-180. doi: 10.1016/j.physa.2015.12.059 [64] Tirunagari S, Poh N, Wells K, et al. Movement correction in DCE-MRI through windowed and reconstruction dynamic mode decomposition[J]. Machine Vision and Applications, 2017, 28(3-4):393-407. doi: 10.1007/s00138-017-0835-5 [65] Sampath R, Chakravarthy S R. Proper orthogonal and dynamic mode decompositions of time-resolved PIV of confined backward-facing step flow[J]. Experiments in Fluids, 2014, 55(9):1792. doi: 10.1007/s00348-014-1792-7 [66] Horchler T, Mani K V, Hannemann K. Dynamic mode decomposition of backward facing step flow simulation data. AIAA 2015-3411[R]. Reston: AIAA, 2015. [67] Ma X Y, Geisler R, Schröder A. Experimental investigation of three-dimensional vortex structures downstream of vortex generators over a backward-facing step[J]. Flow, Turbulence and Combustion, 2017, 98(2):389-415. doi: 10.1007/s10494-016-9768-8 [68] Statnikov V, Roidly B, Meinke M, et al. Analysis of spatio-temporal wake modes of space launchers at transonic flow. AIAA 2016-1116[R]. Reston: AIAA, 2016. [69] Statnikov V, Bolgar I, Scharnowski S, et al. Analysis of characteristic wake flow modes on a generic transonic backward-facing step configuration[J]. European Journal of Mechanics-B/Fluids, 2016, 59:124-134. doi: 10.1016/j.euromechflu.2016.05.008 [70] Statnikov V, Meinke M, Schröder W. Reduced-order analysis of buffet flow of space launchers[J]. Journal of Fluid Mechanics, 2017, 815:1-25. doi: 10.1017/jfm.2017.46 [71] Debesse P, Pastur L, Lusseyran F, et al. A comparison of data reduction techniques for the aeroacoustic analysis of flow over a blunt flat plate[J]. Theoretical and Computational Fluid Dynamics, 2015, 30(3):253-274. doi: 10.1007/s00162-015-0375-4 [72] Seena A, Sung H J. Dynamic mode decomposition of turbulent cavity flows for self-sustained oscillations[J]. International Journal of Heat and Fluid Flow, 2011, 32(6):1098-1110. doi: 10.1016/j.ijheatfluidflow.2011.09.008 [73] Ghommem M, Presho M, Calo V M, et al. Mode decomposition methods for flows in high-contrast porous media. Global-local approach[J]. Journal of Computational Physics, 2013, 253:226-238. doi: 10.1016/j.jcp.2013.06.033 [74] Ferrer E, De Vicente J, Valero E. Low cost 3D global instability analysis and flow sensitivity based on dynamic mode decomposition and high-order numerical tools[J]. International Journal for Numerical Methods in Fluids, 2014, 76(3):169-184. doi: 10.1002/fld.v76.3 [75] Casper K M, Arunajatesan S. Modal decomposition of pressure data in cavity flows. AIAA 2015-2938[R]. Reston: AIAA, 2015. [76] Vinha N, Meseguer-Garrido F, De Vicente J, et al. A dynamic mode decomposition of the saturation process in the open cavity flow[J]. Aerospace Science and Technology, 2016, 52:198-206. doi: 10.1016/j.ast.2016.02.036 [77] Zhang C, Wan Z H, Sun D J. Mode transition and oscillation suppression in supersonic cavity flow[J]. Applied Mathematics and Mechanics, 2016, 37(7):941-956. doi: 10.1007/s10483-016-2095-9 [78] Sampath P, Sinhamahapatra K P. Numerical analysis of characteristic features of shallow and deep cavity in supersonic flow[J]. International Journal of Computational Fluid Dynamics, 2016, 30(3):231-255. doi: 10.1080/10618562.2016.1194976 [79] 叶坤, 叶正寅, 武洁, 等. DMD和POD对超燃冲压发动机凹腔流动的稳定性分析[J].气体物理, 2016, 1(5):39-51. http://industry.wanfangdata.com.cn/dl/Column/Paper?f=detail&q=%e5%85%b3%e9%94%ae%e8%af%8d%3a%22self-sustained+oscillation%22+DBID%3aWF_QKYe K, Ye Z Y, Wu J, et al. Stability analysis of scramjet open cavity flow base on POD and DMD method[J]. Physics of Gases, 2016, 1(5):39-51. (in Chinese) http://industry.wanfangdata.com.cn/dl/Column/Paper?f=detail&q=%e5%85%b3%e9%94%ae%e8%af%8d%3a%22self-sustained+oscillation%22+DBID%3aWF_QK [80] Schmid P J. Application of the dynamic mode decomposition to experimental data[J]. Experiments in Fluids, 2011, 50(4):1123-1130. doi: 10.1007/s00348-010-0911-3 [81] Semeraro O, Bellani G, Lundell F. Analysis of time-resolved PIV measurements of a confined turbulent jet using POD and Koopman modes[J]. Experiments in Fluids, 2012, 53(5):1203-1220. doi: 10.1007/s00348-012-1354-9 [82] Song G, Alizard F, Robinet J C, et al. Global and Koopman modes analysis of sound generation in mixing layers[J]. Physics of Fluids, 2013, 25(12):124101. doi: 10.1063/1.4834438 [83] Kalghatgi P, Acharya S. Modal analysis of inclined film cooling jet flow[J]. Journal of Turbomachinery, 2014, 136(8):081007. doi: 10.1115/1.4026374 [84] Alenius E. Mode switching in a thick orifice jet, an LES and dynamic mode decomposition approach[J]. Computers & Fluids, 2014, 90:101-112. http://cn.bing.com/academic/profile?id=108daaadc0628ed84a6b463be543b866&encoded=0&v=paper_preview&mkt=zh-cn [85] Nair V, Alenius E, Boij S, et al. Inspecting sound sources in an orifice-jet flow using Lagrangian coherent structures[J]. Computers & Fluids, 2016, 140:397-405. https://www.sciencedirect.com/science/article/pii/S0045793016302626 [86] Wan Z H, Zhou L, Wang B F, et al. Dynamic mode decomposition of forced spatially developed transitional jets[J]. European Journal of Mechanics-B/Fluids, 2015, 51:16-26. doi: 10.1016/j.euromechflu.2014.12.001 [87] Chai X C, Iyer P S, Mahesh K. Numerical study of high speed jets in crossflow[J]. Journal of Fluid Mechanics, 2015, 785:152-188. doi: 10.1017/jfm.2015.612 [88] Iyer P S, Mahesh K. A numerical study of shear layer characteristics of low-speed transverse jets[J]. Journal of Fluid Mechanics, 2016, 790:275-307. doi: 10.1017/jfm.2016.7 [89] Hua J C, Gunaratne G H, Talley D G, et al. Dynamic-mode decomposition based analysis of shear coaxial jets with and without transverse acoustic driving[J]. Journal of Fluid Mechanics, 2016, 790:5-32. doi: 10.1017/jfm.2016.2 [90] Jourdain G, Eriksson L-E, Kim S H, et al. Application of dynamic mode decomposition to acoustic-modes identification and damping in a 3-dimensional chamber with baffled injectors[J]. Journal of Sound and Vibration, 2013, 332(18):4308-4323. doi: 10.1016/j.jsv.2013.02.041 [91] Markovich D M, Abdurakipov S S, Chikishev L M, et al. Comparative analysis of low-and high-swirl confined flames and jets by proper orthogonal and dynamic mode decompositions[J]. Physics of Fluids, 2014, 26(6):065109. doi: 10.1063/1.4884915 [92] Motheau E, Nicoud F, Poinsot T. Mixed acoustic-entropy combustion instabilities in gas turbines[J]. Journal of Fluid Mechanics, 2014, 749:542-576. doi: 10.1017/jfm.2014.245 [93] Carlsson H, Carlsson C, Fuchs L, et al. Large eddy simulation and extended dynamic mode decomposition of flow-flame interaction in a lean premixed low swirl stabilized flame[J]. Flow, Turbulence and Combustion, 2014, 93(3):505-519. doi: 10.1007/s10494-014-9560-6 [94] Roy S, Hua J C, Barnhill W, et al. Deconvolution of reacting-flow dynamics using proper orthogonal and dynamic mode decompositions[J]. Physical Review E, 2015, 91(1):013001. doi: 10.1103/PhysRevE.91.013001 [95] Ghani A, Gicquel L, Poinsot T. Acoustic analysis of a liquid fuel swirl combustor using dynamic mode decomposition[R]. ASME Turbo Expo: Turbine Technical Conference and Exposition, 2015, GT2015-42769. [96] Papadogiannis D, Duchaine F, Gicquel L, et al. Assessment of the indirect combustion noise generated in a transonic high-pressure turbine stage[R]. ASME Turbo Expo: Gas Turbine Technical Congress & Exposition, 2015, GT2015-42399. [97] Sampath R, Chakravarthy S R. Investigation of intermittent oscillations in a premixed dump combustor using time-resolved particle image velocimetry[J]. Combustion and Flame, 2016, 172:309-325. doi: 10.1016/j.combustflame.2016.06.018 [98] Roy S, Yi T, Jiang N, et al. Dynamics of robust structures in turbulent swirling reacting flows[J]. Journal of Fluid Mechanics, 2017, 816:554-585. doi: 10.1017/jfm.2017.71 [99] Tu J H, Rowley C W, Aram E, et al. Koopman spectral analysis of separated flow over a finite-thickness flat plate with elliptical leading edge. AIAA 2011-2038[R]. Reston: AIAA, 2011. [100] Ma B F, Liu T X. Low-frequency vortex oscillation around slender bodies at high angles-of-attack[J]. Physics of Fluids, 2014, 26(9):091701. doi: 10.1063/1.4895599 [101] Lee J B, Park S G, Kim B, et al. Heat transfer enhancement by flexible flags clamped vertically in a Poiseuille channel flow[J]. International Journal of Heat and Mass Transfer, 2017, 107:391-402. doi: 10.1016/j.ijheatmasstransfer.2016.11.057 [102] Parkin D J, Thompson M C, Sheridan J. Numerical analysis of bluff body wakes under periodic open-loop control[J]. Journal of Fluid Mechanics, 2014, 739:94-123. doi: 10.1017/jfm.2013.618 [103] Clainche S L, Li J I, Theofilis V, et al. Flow around a hemisphere-cylinder at high angle of attack and low Reynolds number. Part Ⅰ:Experimental and numerical investigation[J]. Aerospace Science and Technology, 2015, 44:77-87. https://www.sciencedirect.com/science/article/pii/S1270963814000844 [104] Clainche S L, Li J I, Theofilis V, et al. Flow around a hemisphere-cylinder at high angle of attack and low Reynolds number. Part Ⅱ:POD and DMD applied to reduced domains[J]. Aerospace Science and Technology, 2015, 44:88-100. doi: 10.1016/j.ast.2014.10.009 [105] Hemati M S, Deem E A, Williams M O, et al. Improving separation control with noise-robust variants of dynamic mode decomposition. AIAA 2016-1103[R]. Reston: AIAA, 2016. [106] Liu Y Z, Zhang Q S. Dynamic mode decomposition of separated flow over a finite blunt plate:time-resolved particle image velocimetry measurements[J]. Experiments in Fluids, 2015, 56(7). doi: 10.1007/s00348-015-2021-8 [107] Lengani D, Simoni D, Ubaldi M, et al. Experimental investigation on the time-space evolution of a laminar separation bubble by proper orthogonal decomposition and dynamic mode decomposition[J]. Journal of Turbomachinery, 2017, 139(3):031006. http://cn.bing.com/academic/profile?id=0be9053f6972fe70d503457a08d5ae7f&encoded=0&v=paper_preview&mkt=zh-cn [108] Kou J, Zhang W, Liu Y, et al. The lowest Reynolds number of vortex-induced vibrations[J]. Physics of Fluids, 2017, 29(4):041701. doi: 10.1063/1.4979966 [109] Jardin T, Bury Y. Lagrangian and spectral analysis of the forced flow past a circular cylinder using pulsed tangential jets[J]. Journal of Fluid Mechanics, 2012, 696:285-300. doi: 10.1017/jfm.2012.35 [110] Bagheri S. Koopman-mode decomposition of the cylinder wake[J]. Journal of Fluid Mechanics, 2013, 726:596-623. doi: 10.1017/jfm.2013.249 [111] He G S, Wang J J, Pan C. Initial growth of a disturbance in a boundary layer influenced by a circular cylinder wake[J]. Journal of Fluid Mechanics, 2013, 718:116-130. doi: 10.1017/jfm.2012.599 [112] Zhang Q S, Liu Y Z, Wang S F. The identification of coherent structures using proper orthogonal decomposition and dynamic mode decomposition[J]. Journal of Fluids and Structures, 2014, 49:53-72. doi: 10.1016/j.jfluidstructs.2014.04.002 [113] Thompson M C, Radi A, Rao A, et al. Low-Reynolds-number wakes of elliptical cylinders:from the circular cylinder to the normal flat plate[J]. Journal of Fluid Mechanics, 2014, 751:570-600. doi: 10.1017/jfm.2014.314 [114] Shaafi K, Vengadesan S. Wall proximity effects on the effectiveness of upstream control rod[J]. Journal of Fluids and Structures, 2014, 49:112-134. doi: 10.1016/j.jfluidstructs.2014.04.005 [115] Ma L Q, Feng L H, Pan C, et al. Fourier mode decomposition of PIV data[J]. Science China Technological Sciences, 2015, 58(11):1935-1948. doi: 10.1007/s11431-015-5908-y [116] Zhang W, Wang Y, Qian Y H. Stability analysis for flow past a cylinder via lattice Boltzmann method and dynamic mode decomposition[J]. Chinese Physics B, 2015, 24(6):064701. doi: 10.1088/1674-1056/24/6/064701 [117] 张宾, 唐湛棋, 孙姣, 等.旋转圆柱绕流的动力模态分析[J].河北工业大学学报, 2015, 44(4):63-67. http://d.old.wanfangdata.com.cn/Periodical/hbgydxxb201504013Zhang B, Tang Z Q, Sun J, et al. Dynamic mode decomposition of flow around rotating cylinder[J]. Journal of Hebei University of Technology, 2015, 44(4):63-67. (in Chinese) http://d.old.wanfangdata.com.cn/Periodical/hbgydxxb201504013 [118] Stankiewicz W, Morzyński M, Kotecki K, et al. Modal decomposition-based global stability analysis for reduced order modeling of 2D and 3D wake flows[J]. International Journal for Numerical Methods in Fluids, 2016, 81(3):178-191. doi: 10.1002/fld.v81.3 [119] Stankiewicz W, Kotecki K, Morzynski M, et al. Model order reduction for a flow past a wall-mounted cylinder[J]. Archives of Mechanics, 2016, 68(2):161-176. http://am.ippt.pan.pl/am/article/view/v68p161 [120] Wang S F, Liu Y Z. Wake dynamics behind a seal-vibrissa-shaped cylinder:a comparative study by time-resolved particle velocimetry measurements[J]. Experiments in Fluids, 2016, 57(3):32. doi: 10.1007/s00348-016-2117-9 [121] Wang L, Feng L H. Extraction and reconstruction of individual vortex-shedding mode from bistable flow[J]. AIAA Journal, 2017, 55(7):2129-2141. doi: 10.2514/1.J055306 [122] Sarkar S, Ganguly S, Dalal A, et al. Mixed convective flow stability of nanofluids past a square cylinder by dynamic mode decomposition[J]. International Journal of Heat and Fluid Flow, 2013, 44(4):624-634. https://www.sciencedirect.com/science/article/pii/S0142727X13001896 [123] Tang Z Q, Jiang N. Dynamic mode decomposition of hairpin vortices generated by a hemisphere protuberance[J]. Science China Physics, Mechanics & Astronomy, 2012, 55(1):118-124. http://phys.scichina.com:8083/Jwk_sciG_en/EN/abstract/abstract505853.shtml [124] Grilli M, Schmid P J, Hickel S, et al. Analysis of unsteady behaviour in shockwave turbulent boundary layer interaction[J]. Journal of Fluid Mechanics, 2012, 700:16-28. doi: 10.1017/jfm.2012.37 [125] Grilli M, Vazquez-Quesada A, Ellero M. Transition to turbulence and mixing in a viscoelastic fluid flowing inside a channel with a periodic array of cylindrical obstacles[J]. Physical Review Letters, 2013, 110(17):174501. doi: 10.1103/PhysRevLett.110.174501 [126] Sarmast S, Dadfar R, Mikkelsen R F, et al. Mutual inductance instability of the tip vortices behind a wind turbine[J]. Journal of Fluid Mechanics, 2014, 755:705-731. doi: 10.1017/jfm.2014.326 [127] Sayadi T, Schmid P J, Nichols J W, et al. Reduced-order representation of near-wall structures in the late transitional boundary layer[J]. Journal of Fluid Mechanics, 2014, 748:278-301. doi: 10.1017/jfm.2014.184 [128] Subbareddy P K, Bartkowicz M D, Candler G V. Direct numerical simulation of high-speed transition due to an isolated roughness element[J]. Journal of Fluid Mechanics, 2014, 748:848-878. doi: 10.1017/jfm.2014.204 [129] Ponitz B, Sastuba M, Brücker C. 4D visualization study of a vortex ring life cycle using modal analyses[J]. Journal of Visualization, 2015, 19(2):237-259. http://cn.bing.com/academic/profile?id=f52f2b46c4fb4f250cfb4fb004dcc491&encoded=0&v=paper_preview&mkt=zh-cn [130] 秦文瑾, 许敏, 孔令逊, 等.动态模态分解方法在缸内湍流场研究中的应用[J].内燃机学报, 2016, 34(4):334-338. http://www.cqvip.com/QK/95968X/201604/669798037.htmlQin W J, Xu M, Kong L X, et al. Application of dynamic mode decomposition in in-cylinder flow field study. Transactions of CSICE, 2016, 34(4):334-338. (in Chinese) http://www.cqvip.com/QK/95968X/201604/669798037.html [131] Priebe S, Tu J H, Rowley C W, et al. Low-frequency dynamics in a shock-induced separated flow[J]. Journal of Fluid Mechanics, 2016, 807:441-477. doi: 10.1017/jfm.2016.557 [132] André T, Durant A, Fedioun I. Numerical Study of supersonic boundary-layer transition due to sonic wall injection[J]. AIAA Journal, 2017, 55(5):1530-1547. doi: 10.2514/1.J055164 [133] Nichols J W, Larsson J, Bernardini M, et al. Stability and modal analysis of shock/boundary layer interactions[J]. Theoretical and Computational Fluid Dynamics, 2017, 31(1):33-50. doi: 10.1007/s00162-016-0397-6 [134] Pan C, Wang J, Wang J J, et al. Dynamics of an unsteady stagnation vortical flow via dynamic mode decomposition analysis[J]. Experiments in Fluids, 2017, 58(3):21. doi: 10.1007/s00348-017-2306-1 [135] 韩忠华, 王绍楠, 韩莉, 等.一种基于动模态分解的翼型流动转捩预测新方法[J].航空学报, 2017, 38(1):30-46. http://d.old.wanfangdata.com.cn/Periodical/hkxb201701003Han Z, Wang S, Han L, et al. A novel method for automatic transition prediction of flows over airfoils based on dynamic mode decomposition[J]. Acta Aeronautica et Astronautica Sinica, 2017, 38(1):30-46. (in Chinese) http://d.old.wanfangdata.com.cn/Periodical/hkxb201701003 [136] 洪树立, 黄国平.引入DMD方法研究有/无控气流分离的动态结构[J].航空学报, 2017, 38(8):120876. http://mall.cnki.net/magazine/Article/HKXB201708002.htmHong S L, Huang G P. Introduce DMD method to study the dynamic structures of flow separation with and without control[J]. Acta Aeronautica et Astronautica Sinica, 2017, 38(8):120876. (in Chinese) http://mall.cnki.net/magazine/Article/HKXB201708002.htm [137] Mariappan S, Gardner A D, Richter K, et al. Analysis of dynamic stall using dynamic mode decomposition technique[J]. AIAA Journal, 2014, 52(11):2427-2439. doi: 10.2514/1.J052858 [138] Mohan A T, Gaitonde D V, Visbal M R. Model reduction and analysis of deep dynamic stall on a plunging airfoil using dynamic mode decomposition[R]. AIAA 2015-1058. [139] Mohan A T, Agostini L, Gaitonde D V, et al. A preliminary spectral decomposition and scale separation analysis of a high-fidelity dynamic stall dataset. AIAA 2016-1352[R]. Reston: AIAA, 2016. [140] 潘翀, 陈皇, 王晋军. 复杂流场的动力学模态分解[C]//第八届全国实验流体力学学术会议论文集. 广州: 中国科学院南海海洋研究所, 2010: 77-82.Pan C, Chen H, Wang J J. Dynamical mode decomposition of complex flow field[C]//8th National Conference on Experimental Fluid Mechanics, Guangzhou: South China Sea Institute of Oceanology, 2010: 77-82. (in Chinese) [141] Pan C, Yu D S, Wang J J. Dynamical mode decomposition of Gurney flap wake flow[J]. Theoretical & Applied Mechanics Letters, 2011, 1(1):42-012002. https://www.sciencedirect.com/science/article/pii/S2095034915300118 [142] Ducoin A, Loiseau J C, Robinet J C. Numerical investigation of the interaction between laminar to turbulent transition and the wake of an airfoil[J]. European Journal of Mechanics-B/Fluids, 2016, 57:231-248. doi: 10.1016/j.euromechflu.2016.01.005 [143] Dawson S T M, Floryan D C, Rowley C W, et al. Lift enhancement of high angle of attack airfoils using periodic pitching. AIAA 2016-2069[R]. Reston: AIAA, 2016. [144] Dawson S T M, Schiavone N K, Rowley C W, et al. A data-driven modeling framework for predicting forces and pressures on a rapidly pitching airfoil[R]. AIAA 2015-2767. [145] 寇家庆, 张伟伟, 高传强.基于POD和DMD方法的跨声速抖振模态分析[J].航空学报, 2016, 37(9):2679-2689. https://www.researchgate.net/profile/Jiaqing_Kou/publication/289973603_Modal_Analysis_of_Transonic_Buffet_Based_on_POD_and_DMD_Techniques_in_Chinese/links/5693c01908aeab58a9a2a81b/Modal-Analysis-of-Transonic-Buffet-Based-on-POD-and-DMD-Techniques-in-Chinese.pdfKou J Q, Zhang W W, Gao C Q. Modal analysis of transonic buffet based on POD and DMD method[J]. Acta Aeronautica et Astronautica Sinica, 2016, 37(9):2679-2689. (in Chinese) https://www.researchgate.net/profile/Jiaqing_Kou/publication/289973603_Modal_Analysis_of_Transonic_Buffet_Based_on_POD_and_DMD_Techniques_in_Chinese/links/5693c01908aeab58a9a2a81b/Modal-Analysis-of-Transonic-Buffet-Based-on-POD-and-DMD-Techniques-in-Chinese.pdf [146] Kim J, Moin P, Seifert A. LES-based characterization of a suction and oscillatory blowing fluidic actuator. AIAA 2016-0572[R]. Reston: AIAA, 2016. [147] Reagan A J, Dubief Y, Dodds P S, et al. Predicting flow reversals in a computational fluid dynamics simulated thermosyphon using data assimilation[J]. PLoS ONE, 2016, 11(2):e0148134. doi: 10.1371/journal.pone.0148134 [148] Muld T W, Efraimsson G, Henningson D S. Flow structures around a high-speed train extracted using proper orthogonal decomposition and dynamic mode decomposition[J]. Computers & Fluids, 2012, 57:87-97. https://www.sciencedirect.com/science/article/pii/S0045793011003847 [149] 丁杰, 胡晨星, 刘鹏寅, 等.无叶扩压器中非定常流动的DMD模态分析[J].动力工程学报, 2017, 37(6):447-453. http://manu31.magtech.com.cn/Jwk_dlgcxb/CN/volumn/volumn_1197.shtmlDing J, Hu C X, Liu P Y, et al. Dynamic mode decomposition of the unsteady flow in a vaneless diffuser[J]. Journal of Chinese Society of Power Engineering, 2017, 37(6):447-453. (in Chinese) http://manu31.magtech.com.cn/Jwk_dlgcxb/CN/volumn/volumn_1197.shtml [150] Brunton S L, Brunton B W, Proctor J L, et al. Chaos as an intermittently forced linear system[J]. Nature Communications, 2017, 8:19. doi: 10.1038/s41467-017-00030-8 [151] Gao C Q, Zhang W W, Ye Z Y. Numerical study on closed-loop control of transonic buffet suppression by trailing edge flap[J]. Computers & Fluids, 2016, 132:32-45. https://www.sciencedirect.com/science/article/pii/S0045793016300949 [152] Noack B R, Afanasiev K, Morzyński M, et al. A hierarchy of low-dimensional models for the transient and post-transient cylinder wake[J]. Journal of Fluid Mechanics, 2003, 497:335-363. doi: 10.1017/S0022112003006694 [153] 尹明朗, 寇家庆, 张伟伟.一种高泛化能力的神经网络气动力降阶模型[J].空气动力学学报, 2017, 35(2):205-213. http://www.kqdlxxb.com/CN/abstract/abstract12002.shtmlYin M L, Kou J Q, Zhang W W. A reduced-order aerodynamic model with high generalization capability based on neural network[J]. Acta Aerodynamica Sinica, 2017, 35(2):205-213. (in Chinese) http://www.kqdlxxb.com/CN/abstract/abstract12002.shtml -
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