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Open Seminar at Tokyo Polytechnic University POD AND NEW INSIGHTS IN WIND ENGINEERING LE THAI HOA Vietnam National University, Hanoi PART 1 CONTENTS Introduction POD and its Proper Transformations in Time Domain and Frequency Domain New Insights in Wind Engineering Topic 1: POD and Pressure Fields Topic 2: POD and Wind Fields, Wind Simulation Topic 3: POD and Response Prediction Topic 4: POD and System Identification Further Perspectives and Development BRIEF PROFILE Global COE Associate Professor Global Center of Excellence (GCOE) Program Wind Engineering Research Center (WERC) Tokyo Polytechnic University (TPU) 1583 Iiyama, Atsugi, Kanagawa, Japan, 243-0297 Email. [email protected] On the temporary leave from Vietnam National University, Hanoi (VNU) College of Technology (COLTECH) Faculty of Engineering Mechanics 144 Xuanthuy, Caugiay, Hanoi Email. [email protected] • Education • Working Experience • Research Experience • Awards Research Experience and Interests Wind-induced vibrations of civil structures with emphasis on aeroelastic stability analysis (Flutter Instability); Gust response prediction (Buffeting Response) of structures; Cable aerodynamics Wind-resistant design of structures with coding and specification; wind tunnel tests Proper Orthogonal Decomposition and its Proper Transformations with applicable for analysis, simulation, response prediction and system identification of wind effects on structures Time – Frequency Analysis (TFA) and its Applications with applicable for analysis, simulation, response prediction and system identification of wind effects on structures Structural Health Monitoring and Assessments INTRODUCTION (1) Proper Orthogonal Decomposition (POD), known as some other names as Principal Component Analysis (PCA)and the Karhunen-Loeve Decomposition (KLD), has been applied popularly in many engineering fields and in the wind engineering as well. Mathematically, the POD is actually matrix decomposition using eigen problems with concepts of eigenvalues and eigenvectors. But used for either correlated or non-correlated multivariate random data Main advantages are that analysis and synthesis of multivariate random data through simplified (reduced-order) description of few number of fundamental low-order eigenvalues associated with their eigenvectors. INTRODUCTION (2) Multivariate random data can be reorganized and represented under the matrix forms, then are decomposed using the POD. It isNormally, noted that multivariate random data in the wind either zero-time-lag covariance matrix or cross spectral matrix of the multivariate random data such are used, engineering (mostly correlated data) are many as corresponding to the time domain and the frequency domain turbulent wind fields, surface pressure fields, aerodynamic formulations forces and so on, for which the POD can be applicable. The POD and its Proper Transformations have been branched by either the Covariance Proper Transformation (CPT) in time domain or the Spectral Proper Transformation (SPT) in the frequency domain. However, only the CPT has been widely used so far in the wind engineering to some extent. Objectives of Research (1) Better understanding about POD and its recent applications for the wind engineering topics (2) Some new insights of the POD’s recent applications in the wind engineering topics, concretely as Analysis and Synthesis of Pressure Fields; Digital Simulation of Turbulent Wind Fields; Stochastic Response Prediction of Structures; and System Identification of Structures Under the lights of both time domain and frequency domain. POD AND ITS TRANFORMATIONS IN TIME DOMAIN AND FREQUENCY DOMAIN Eigenvalue-based Matrix Decompositions Proper Orthogonal Decomposition (POD) Covariance Matrix and Cross Spectral Matrix Covariance Proper Transformation (CPT) Spectral Proper Transformation (SPT) Recent Applications of POD Eigenvalue-based Matrix Decomposition Since the multivariate random data can be represented under the matrix form, the matrix decomposition techniques can be exploited, concretely the eigenvalue-based matrix decompositions used. Decomposition Forward Eigenvalue-based Matrix Decomposition Field Backward Reconstruction Proper Orthogonal Schur Singular Value Eigen Decomposition Decomposition(SD) Decomposition(SVD) Decomposition(POD) (ED) A VDV T A: real, square V: orthogonal A VUV *T A: complex, square V: unitary A VDV *T A SDV *T A: complex, rectangular A: complex, square S, V: unitary V: unitary Fig. 1 Eigenvalue-based matrix decomposition POD Proper Orthogonal Decomposition (POD) is as eigenvaluebased (orthogonal) matrix decomposition methods. Then, the matrix is approximated in the reduced-order model based in its eigenvalues and eigenvectors. POD is considered as mathematical tool (eigenvalue-based) used to decompose and approximate the random fields under more simplified ways; low-dimensional approximate description of high-dimensional process. POD branched by (1) Covariance Proper Transformation based on covariance matrix (2) Spectral Proper Transformation based on cross spectral matrix Overall Overviews Actually, the POD has been developed by several people. Principal Component Analysis (PCA) firstly introduced by Pearson (1901), Hotelling (1933) Karhunen-Loeve Decomposition (KLD) by Loeve (1945) and Karhunen (1946) and others POD might be named by Lumley (1970), Holmes and Lumley (1996) with first application for studying turbulence and coherence structures in fluid media POD and wind engineering (pressure fields) have pioneered by Holmes (1987,1990), Bienkiewicz (1995), Tamura (1997, 1999, 2001) Applications of the POD in the wind engineering still are evolving Matrix Representation of Random Fields Multi-variate random fields (wind velocities, pressure, force…) consisting of N-point time series) are represented comprehensively using matrix forms of either Covariance Matrix or Cross Spectral Matrix For example: (t ) {1 (t ),2 (t ),... , N (t )} T 2(t) 4(t)… 1(t) 3(t) 5(t)… Covariance Matrix R11 (0) R12 (0) R (0) R (0) 2 2 R (0) 2 1 R N1 (0) R N2 (0) Rmk (0) E m (t )k (t )T R1 N (0) R2 N (0) R N N (0) m (t ),k (t ) : Pressure time series E[] : Expectation value Surface Pressure field Body Cross Spectral Matrix S11 (n) S12 (n) S ( n) S ( n) 2 2 S (n) 2 1 S N1 (n) S N2 (n) S1 N (n) S2 N (n) S N N (n) Smk (n) Smm (n) Skk (n)COH (n, mk ) Smm (n), Skk (n) : Auto spectral elements COH (n, mk ) : Coherent function n : Frequency variable Covariance Proper Transformation (CPT) Covariance matrix-based POD find out pairs of the covariance eigenvalues and orthogonal eigenvectors: R (0) where diag[ 1 , 2 ,... N ] : Covariance eigenvalues : Covariance eigenvectors [1 , 2 ,...N ] R (0) : Zero-time-lag covariance matrix Covariance Proper Transformation (CPT): approximation of the turbulent fields: ~ M (t ) x (t ) j xj (t ) j 1 ~ M ( N ) :Number of covariance modes; x(t): covariance where N 1 principal coordinate: X (t ) (t ) (t ) i (t )i i 1 Spectral Proper Transformation (SPT) Similarly, the cross spectral matrix-based POD is to find out pairs of spectral eigenvalues and eigenvectors: S (n) (n) (n) (n) where (n) diag[1 (n), 2 (n),...N (n)] : Spectral eigenvalues (n) [ 1 (n), 2 (n),... N (n)] : Spectral eigenvector S (n) : Cross spectral matrix (t ); u; w Spectral Proper Transformation (SPT) : approximation of power spectral density functions Mˆ S (n) (n) (n)*T (n) j (n)j (n) *jT (n) j 1 where Mˆ ( N: ): Number of spectral turbulent modes Mˆ ˆ (n) (n)Y (n) (n) y (n) Y (n): Spectral principal coordinate j 1 i i xˆ (t ) (n) yˆ (n) exp(i 2nt)dn 0 Relationship between CPT and SPT (n) (n)*T (n)dn T 0 Relationships between the CPT and the SPT can be expressed as follows (From forward and backward Fourier transform in the first-order and second-order) First-order relationship: between covariance and spectral principal coordinates X (t ) Y (n) exp( i 2nt )dn 0 X (t ) (n)Y (n) exp( i 2nt )dn 0 Second-order relationship: between covariance matrix and cross spectral matrix R S (n)dn 0 (n) (n) (n)*T dn T 0 Recent Applications of POD In the wind engineering topics In Frequency domain In Time domain POD New lines and new insights Analysis & Synthesis Digital Simulation Stochastic Response System Identification of Structures of Structures of Pressure Fields of Turbulent Fields In frequency domains In time domains Shinuzoka(1991), Holmes 1990, Tamura (1997,1999) Di Paola (2001) … … In frequency & time domains Carrasale(1999), Solari (2007) … Fig. 2 POD applications in the wind engineering topics NEW INSIGHTS IN WIND ENGINEERING TOPIC 1 : POD and Analysis, Synthesis and Identification of Unsteady Pressure Field TOPIC 2 : POD and Digital Simulation of Turbulent Wind Field, Understanding Turbulent Field TOPIC 3 : POD and Stochastic Response Prediction of Wind-excited Structures TOPIC 4 : POD and System Identification of Wind-excited Structures TOPIC 1 ANALYSIS, SYNTHESIS AND IDENTIFICATION OF UNSTEADY PRESSURE FIELDS IN TIME DOMAIN AND FREQUENCY DOMAIN Introduction Experimental set-ups Covariance matrix-based POD analysis Spectral matrix-based POD analysis Identification of pressure fields and physical linkage Results and discussions Remarks and New Insights Introduction POD has applied long stance for analyzing and identifying physical pressure fields (Holmes et al. 1988, 1997, Bienkiewicz et al.1995, Tamura et al. 1997,1999,2001). Linkage between POD modes and physical causes is usually looked Both covariance matrix-based POD in the time domain for to establish. and spectral matrix-based in the frequency domain Obviously, it has its advantagePOD to decompose and simplify the will be used. pressure fields. Linkage between the POD modes and the physical So far, all applications of POD for pressure fields is based phenomena on models will be investigated. on covariance matrix-based POD analysis. However, some literatures quoted that this physical linkages are misleading, probably fictitious in many cases due mathematical nature and sensitive constraint of POD (Armit 1967, Holme 1997, Tamura 1999). Questionary on physical meaning “… there is no reason to suppose that spatial variation of the pressure fluctuations due to one physical cause are necessarily orthogonal with respect to that due to another cause. The mathematical constraints caused by orthogonality condition could therefore mean that in some cases, a unique physical cause cannot be associated with each eigenvector.” Armitt, J. of 1968 Not accurate interpretation of physical meaning covariance modes might be modes comeare from: “… the shapes of the constrained by the requirement of orthogonality, and hence any physical interpretation of these modes could be at least misleading, and (1) Number of pressure taps probably fictitious in many cases. The most useful aspect of the proper orthogonal (2) Tap arrangement oreconomical non-uniform) decomposition techniques is(uniform that it is an form for describing the spatial and wind variations on a buildings, or other bluff body, and is (3) temporal Presence of pressure mean pressures especially useful for relating the pressures to structural load effects.” (4) Turbulent conditions Holmes, J.D. 1997 (5) Complexity of bluff body flow, geometry of models (6) And soand on.wrong interpretation of the covariance modes due to presence of “… distortion mean pressure data in the analyzed pressure field.” Tamura, Y. 1997, 1999 Experimental Set-ups Chordwise pressures on some typical rectangular cylinders have been measured in some turbulent flows in wind tunnel (Structure and Wind Engineering Laboratory, Kyoto University) B/D=1 B/D=1 with Splitter Plate B/D=5 B/D=1 with S.P B/D=1 Wind Wind B/D=5 Wind Splitter Plate (S.P) po1… po1… po10 po10 po1… Fig. 3 Physical models: B/D=1, B/D=1 with S.P, B/D=5 B/D=1 B/D=1 with S.P B/D=5 Wind Wind Wind U=3m/s U=6m/s U=9m/s Fig. 4 Flow pattern around models: B/D=1, B/D=1 with S.P, B/D=5 po19 Power Spectral Densities (PSD) of Pressures 10 10 4 10 U=3m/s 3 1.22Hz 10 10 10 10 2 10 Karman Vortex shed in wake at 4.15Hz 0 10 -1 po.1 po.3 po.5 po.7 po.9 -2 10 1 10 po1 po3 po5 po7 po9 0 -1 10 0 10 1 10 10 -1 10 2 10 0 1 10 10 -1 10 2 10 10 10 1 10 2 U=9m/s 2 10 3 3 1 PSD 2 PSD 0 po1 po3 po5 po7 po9 -2 10 0 10 1 10 10 -1 10 2 10 po1 po3 po5 po7 po9 0 -1 10 0 10 10 10 10 -3 -4 po.1 po.2 po.5 po.6 po.9 po.10 po.18 po.19 -5 -6 -1 1 10 10 -1 10 2 10 0 -1 10 10 10 2.44Hz 10 -3 -4 1 10 2 -1 3.42Hz U=9m/s 4.88Hz 7.32Hz -2 10 Frequency n(Hz) po.1 po.2 po.5 po.6 po.9 po.10 po.18 po.19 10 10 6.84Hz -2 -3 -4 po.1 po.2 po.5 po.6 po.9 po.10 po.18 po.19 Vortices shed at reattachment point 10 10 -7 10 10 PSD of pressure(1/Hz) 10 po1 po3 po5 po7 po9 0 10 U=6m/s 2.44Hz PSD of pressure(1/Hz) 10 10 1.22Hz -2 1 Frequency n(Hz) -1 U=3m/s 10 2 1 Frequency n(Hz) 10 10 No Karman Vortex occurred -1 -3 PSD of pressure(1/Hz) 0 4 U=6m/s 10 10 10 Frequency n(Hz) 4 1.22Hz 10 PSD B/D=1 with S.P 10 10 -1 10 B/D=5 10 10 10 po1 po3 po5 po7 po9 1 Frequency n(Hz) 3 U=3m/s 2 0 10 Frequency n(Hz) 10 3 PSD 10 1 -3 10 12.94Hz 4 2 10 -1 10 10 5 U=9m/s 3 PSD PSD B/D=1 10 10 8.79Hz U=6m/s 10 10 4 4.15Hz 10 0 10 Frequency (Hz) 1 10 2 10 -5 -6 10 -7 10 -1 10 10 0 10 Frequency (Hz) 1 10 2 10 -5 -6 -7 10 -1 10 0 10 Frequency (Hz) Fig. 5 Power spectral densities (PSD) of fluctuating pressures 1 10 2 Covariance Matrix-based POD Analysis (1) B/D=1 with S.P B/D=1 B/D=5 Modes Modes Modes 0.8 0.6 1 0.6 0.75 0.4 Table 1: Energy contribution of covariance POD modes (unit: %) 0.4 0.5 0.2 0.25 0 0 -0.2 -0.25 -0.4 -0.5 0.2 0 -0.2 -0.4 mode 1 mode 2 mode 3 mode 4 -0.6 -0.8 1 2 3 4 5 6 Positions 7 8 9 mode 1 mode 2 mode 3 mode 4 -0.75 10 -1 1 2 3 4 5 6 Positions 7 8 9 mode 1 mode 2 mode 3 mode 4 -0.6 10 -0.8 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Positions Fig. 6 First four covariance modes at different physical models Tab. 1 Energy contribution of covariance modes Mode B/D=1 B/D=1 with S.P B/D=5 3m/s 6m/s 9m/s 3m/s 6m/s 9m/s 3m/s 6m/s 9m/s 1 76.92 77.46 75.36 65.29 62.79 63.30 43.77 44.86 65.9 2 13.27 13.25 14.41 20.97 22.61 22.08 22.02 23.14 13.29 3 4.69 4.23 4.62 6.14 6.29 6.10 15.18 15.14 9.48 4 2.87 2.86 3.17 4.04 4.32 4.41 5.98 5.68 3.4 5 1.27 1.32 1.45 1.99 2.28 2.45 4.76 4.11 2.79 Covariance Matrix-based POD Analysis (2) B/D=1 with S.P B/D=1 Coordinate 1 Coordinate 2 Coordinate 1 B/D=5 Coordinate 1 Coordinate 2 Coordinate 2 20 20 10 10 10 10 10 10 5 5 5 5 0 0 0 0 0 0 -10 -10 -5 -5 -5 -5 -20 0 20 5 Time (s) 10 Coordinate 3 -20 0 5 Time (s) -10 0 10 Coordinate 4 20 10 5 Time (s) 10 Coordinate 3 -10 0 5 Time (s) -10 0 10 Coordinate 4 10 10 5 Time (s) 10 Coordinate 3 -10 0 10 5 5 5 5 0 0 0 0 0 0 -10 -10 -5 -5 -5 -5 5 Time (s) 10 -20 0 5 Time (s) -10 0 10 5 Time (s) 10 -10 0 5 Time (s) f: Karman vortex 10 10 10 4.15Hz 0 10 -1 10 -2 PSD 10 10 10 -3 -4 -5 -6 10 -1 10 5 Time (s) 10 f: vortex shedding 5 Time (s) 10 1.22Hz 0 10 -1 10 10 Principal coordinates 1 1 -10 0 coordinate 1 coordinate 2 coordinate 3 coordinate 4 0 -1 First covariance principal coordinates contains frequency peaks of physical causes 10 PSD 10 1 10 10 coordinate 1 coordinate 2 coordinate 3 coordinate 4 10 -2 -3 0 10 Frequency (Hz) 1 10 2 10 -4 -5 -6 10 10 PSD 10 -10 0 10 10 Coordinate 4 10 10 -20 0 5 Time (s) 10 -1 10 10 coordinate 1 coordinate 2 coordinate 3 coordinate 4 10 -2 -3 -4 -5 -6 10 0 10 1 10 2 10 -1 10 Frequency (Hz) Fig. 7 Covariance principal coordinates and their PSD 10 0 10 Frequency (Hz) 1 10 2 Covariance Matrix-based POD Analysis (3) B/D=1 B/D=1 with S.P target 2rd mode 0 500 1000 1500 2000 8 6 4 2 0 -2 -4 -6 -8 1000 10 PSD 10 10 PSD -3 10 10 10 target 10 0 10 10 1st mode -1 10 1 10 2 10 10 0 -3 10 10 10 -5 target 10 3rd mode 10 -1 10 0 1000 10 Frequency (Hz) 1 10 2 1500 2000 2000 -3 target 10 1 10 10 1500 2000 -3 10 0 10 1 10 2 10 target 10 1000 10 Frequency (Hz) 10 2 10 10 -5 target 10 2000 -1 10 0 10 10 Frequency (Hz) 1 10 2 0 500 1000 1500 Amplitude 6 target 4st modes 4 0 -2 10 10 -3 10 2000 Position 5 2 0 -2 -4 0 500 1000 1500 -6 2000 0 500 1000 Time (ms) 0 Position 5 1 1500 2000 Time (ms) 10 0 10 -3 10 Position 5 1 0 -3 -5 target -1 10 10 0 10 1 10 2 10 10 -3 10 target -5 1st mode 10 Position 5 1 3rd mode 1 1500 10 0 10 -3 -6 2000 2 Position 5 1 10 Position 5 4st mode 10 500 2nd mode -5 0 0 1st mode 10 10 10 10 target 1 -3 -1 -2 10 10 0 10 -7 0 10 -1 1500 2rd mode Time (ms) 0 -5 1000 target 2 Position 5 1 500 4 -6 1000 0 -2 6 4st modes -4 500 0 target 4 -6 0 2 Position 5 -4 2 0 2000 -6 Position 5 2 1500 -4 10 2nd mode 10 1000 6 2rd mode -2 10 0 500 Time (ms) 0 -1 0 2nd mode -4 Position 5 0 10 -5 Amplitude Amplitude 500 2 Position 5 2 10 Position 5 2 0 target PSD 0 -5 -6 4 PSD 10 10 PSD PSD 10 -4 Time (ms) Position 5 2 1500 -2 -6 6 500 0 -4 4st modes 0 2 -6 target Time (ms) 10 -2 Position 5 Amplitude 8 6 4 2 0 -2 -4 -6 -8 2000 0 -4 Position 5 Amplitude Amplitude Position 5 1500 -2 2 Amplitude 1000 0 target 4 1st mode PSD 500 2 4 2nd mode Amplitude 0 4 1st mode PSD 2000 4 Position 5 6 target PSD 1500 2nd mode Amplitude 1000 Position 5 6 target PSD 500 Position 5 6 target PSD 0 Position 5 6 target Amplitude 1st mode 8 6 4 2 0 -2 -4 -6 -8 -1 10 0 10 10 1 10 2 Position 5 1 target -5 10 2nd mode -1 10 0 10 1 10 2 Position 5 1 10 0 10 0 PSD Position 5 target Amplitude Amplitude Position 5 8 6 4 2 0 -2 -4 -6 -8 B/D=5 -3 10 -3 -5 target -7 10 10 4st mode -1 10 0 10 Frequency (Hz) 1 10 2 -5 10 target -5 target 3rd mode 10 -1 10 0 4st mode 10 1 10 2 Frequency (Hz) Fig. 8 Contribution of covariance modes on original pressures 10 -1 10 0 10 Frequency (Hz) 1 10 2 Spectral Matrix-based POD Analysis (1) B/D=1 10 Spectral eigenvalues 1 10 0 10 Spectral eigenvalues 1 10 0 10 B/D=5 B/D=1 with S.P -1 10 10 -1 10 -2 10 10 -3 10 10 -5 10 0 10 1 10 2 10 -4 10 -1 10 Frequency (Hz) -1 -2 -3 f: vortex shedding -4 10 0 10 1 10 2 10 -1 10 Frequency (Hz) 10 0 10 1 Frequency (Hz) Fig. 9 First five spectral eigenvalues Tab. 2 Energy contribution of spectral modes Mode -3 -5 10 -1 10 -2 f: Karman vortex -4 10 10 0 10 10 Spectral eigenvalues 1 B/D=1 with S.P B/D=1 3m/s 6m/s 9m/s 3m/s 1 86.04 85.84 83.02 2 8.08 8.08 3 3.28 4 5 6m/s 9m/s B/D=5 3m/s 6m/s 9m/s 81.30 77.48 77.88 74.77 73.59 83.93 9.92 10.15 12.36 11.98 12.68 14.03 7.69 3.20 3.68 4.44 5.14 5.00 5.68 5.56 3.57 1.40 1.62 1.94 2.05 2.63 2.70 2.75 2.86 1.86 0.64 0.72 0.81 1.09 1.28 1.34 1.44 1.45 1.06 10 2 Physical meaning of these spectral modes are still unknown B/D=5 B/D=1 with S.P B/D=1 Spectral Matrix-based POD Analysis (2) Fig. 10 First three spectral modes Spectral Matrix-based POD Analysis (3) B/D=1 10 PSD 10 10 10 10 10 10 -1 10 -2 10 -3 10 -4 -5 -6 -7 -8 10 -1 10 10 10 target 1st mode 2nd mode 3rd mode 4th mode 10 10 0 10 1 10 10 -1 10 -2 10 -3 10 -4 -5 -6 -7 10 -1 10 2 Position 5 0 -8 10 target 1st mode 2nd mode 3rd mode 4th mode 10 10 10 10 0 10 1 10 10 -1 10 -2 10 -3 -4 10 10 -5 -6 10 -1 10 10 target 1st mode 1st to 2nd modes 10 Position 5 0 10 Frequency (Hz) -5 -6 -7 target 1st mode 2nd mode 3rd mode 4th mode 10 0 1 10 2 10 1 10 2 10 Position 5 0 -1 10 -1 -2 -3 -2 -3 -4 10 -5 -6 0 -4 PSD PSD PSD 10 -3 Frequency (Hz) 10 10 -2 10 -1 10 2 10 10 -1 -8 10 Position 5 0 Frequency (Hz) Position 5 0 10 10 Frequency (Hz) 10 B/D=5 PSD 10 Position 5 0 PSD 10 B/D=1 with S.P 10 -1 10 target 1st mode 1st to 2nd modes 10 -4 -5 0 10 Frequency (Hz) 1 10 2 10 -1 10 target 1st mode 1st to 2nd modes 10 0 10 Frequency (Hz) Fig. 11 Contribution of spectral mode on PSD of original pressure 1 10 2 Remarks and Insights The first covariance mode and the first spectral mode play very significant role which can characterize for whole pressure field. Concretely, the first covariance mode, the first spectral one contain certainPOD spectral peaksof of the hidden physical Thus, Spectral Matrix-based Analysis surface events, moreover, it contributes dominantly on the field pressure fields with emphasis on investigation of physical energy. meaning of themode spectral modes bethan newthe linecovariance in the POD’s The spectral exhibits the will better applications mode in the synthesis (reconstruction) of the pressure field. Therefore, to some extent only the first mode is accuracy enough to reconstruct and identify the whole pressure field. The linkage between the POD modes and the physical causes has been found out in some investigated cases. However, more investigation should be needed to clarify physical meaning of spectral modes. TOPIC 2 DIGITAL SIMULATION OF MULTIVARIATE TURBULENT WIND FIELDS & UNDERSTANDING TURBULENT FIEDLS Introduction Cholesky Decomposition-based Simulation POD-based Simulation Numerical Examples And Discussions Investigations on Turbulent Wind Fields Remarks and New Insights Introduction Time series simulation of turbulent field have been required as a must in many cases, especially in the time domain analysis. Simulation of correlated multivariate stationary random fields as turbulent field is in difficulty. Generally, simulation methods have been branched by either frequency domain representation or time series parametric ones, but both of them based on decomposition of spectral matrix form of multivariate turbulent fields. Cholesky’s Decomposition Spectral Representation (Indirect Simulation) POD Turbulence Simulation on Auto-Regressive (AR) Cross Spectral ARMA Matrix Time-series Representation (Direct Simulation) All based Fig. 12 Classifications of time-series simulation methods S (n) H (n)H (n)*T Cholesky Decomposition-based Simulation nup nkl The Cholesky decomposition is the most common technique for simulating the multivariate turbulent wind field in thefrequency domain in which the cross spectral matrix is decomposed: S (n) H (n) H (n)*T H (n) : complex lower triangular matrix The multivariate turbulent wind field can be expressed in the frequency domain using the factorized lower triangle matrix j i (t ) 2n k 1 ~ N | H ( n l 1 kl j k ) | cos( 2nkl t jl (nkl ) kl ) where j: index of structural node; k: index of moving node; l: index of moving point nup ~ n in frequency range; N: number of frequency intervals; n: frequency interval ~ N nup : upper cutoff frequency; nkl : frequency point on frequency range n (l 1)n kn / N kl H (nkl ) : element of complex lower-triangle matrix; jl (nkl ) : complex phase angle of H (nkl ) kl : random phase angles, uniformly distributed over [0,2] which are generated by Monte Carlo technique j k j k POD-based Simulation The i-th subprocess in the N-variate spatially-correlated turbulent field can be simulated using the spectral modes: Mˆ i (t ) 2 j 1 Nˆ | (n ) | l 1 j l j (nl )nl cos( 2nl t j (nl ) l ) where l: index of frequency point, nl: frequency at moving point l; Nˆ : number of frequency intervals; nl: frequency interval at l; j (nl ):phase angle of complex eigenvector j;(nl ) l : phase angle as random variable uniformly distributed over [0, 2π] generated by Monte Carlo technique l (nl ) tan 1 Im( j (nl )) / Re( j (nl )) Numerical examples, results and discussions In this numerical example, the spectral proper transformation has been applied to simulate the two multivariate turbulent fields at 30 discrete nodes along line-like structure. The turbulent wind fields are simulated at different mean velocities U=5,10,20,30 and 40m/s with sampling rate of 1000Hz for interval of 100 seconds. u (t ) u1 (t ), u2 (t ),...u30 (t ) T 1 2 i T j 29 ui(t) z uj(t) y w(t ) w1 (t ), w2 (t ),... w30 (t ) Li w wi(t) 30 wj(t) x u Fig. 13 Turbulent fields at line-like structure’s discrete nodes Spectral Eigenvalues u-turbulence w-turbulence 3000 0 -2 10 23.12% 50 40 30 20 17.36% 10 10 -1 10 Frequency n(Hz) 0.5Hz Eigenvalue of Sw(n) 1000 500 60 42.19% 1500 70 0.2Hz Eigenvalue of Su(n) 2500 2000 80 0 10 1 0 -2 10 13.42% 10 -1 10 0 Frequency n(Hz) Fig. 14 First five spectral eigenvalues on spectral band 0-10Hz at U=20m/s 10 1 Spectral Eigenvectors (Modes) form ModeSymmetrical 1 Asymmetrical form Mode 2 form u-turbulence ModeSymmetrical 3 Mode 1 Mode 2 Mode 3 w-turbulence Fig. 15 First three turbulent modes at spectral band 010Hz at U=20m/s Physical Meanings of Turbulent Modes (1) w-turbulent spectra modes U=5m/s U=10m/s U=20m/s U=30m/s U=40m/s No difference in shape and value among turbulent modes at investigated spectral bands Fig.16 Effect of different mean velocities on turbulent modes Physical Meanings of Turbulent Modes (2) Frequency-dependant eigenvalues (w-turbulence) U=20m/s U=5m/s U=40m/s w-turbulence w-turbulence w-turbulence 80 12 10 70 180 160 Eigenvalues express ‘energy contribution’ of turbulent modes 50 140 Eigenvalue of Sw(n) Eigenvalue of Sw(n) Eigenvalue of Sw(n) 8 60 120 100 Turbulent modes do not change at low frequency ranges 6 4 40 30 20 80 60 40 Turbulent modes exhibit as symmetrical or asymmetrical waves (similarly as structural modes). Thus, they can behavior either exciting or suppressing to structural modes. Turbulent modes and associated eigenvalues are supposed to interpret scale and frequency of turbulent eddies of turbulent fields. 2 0 -2 10 10 10 -1 10 0 10 0 -2 10 1 20 10 -1 10 0 10 0 -2 10 1 10 -1 Frequency n(Hz) Frequency n(Hz) Mode 1 Mode 2 180 160 140 0 10 1 Mode 3 40 5m/s 10m/s 20m/s 30m/s 40m/s 10 Frequency n(Hz) 25 5m/s 10m/s 20m/s 30m/s 40m/s 35 30 5m/s 10m/s 20m/s 30m/s 40m/s 20 100 80 25 20 15 60 40 10 20 5 0 -2 10 Eigenvalue Eigenvalue Eigenvalue 120 15 10 5 10 -1 10 Frequency (Hz) 0 10 1 0 -2 10 10 -1 10 0 10 1 0 -2 10 Frequency (Hz) Fig. 17 Effect of different mean velocities on eigenvalues Comparison between eigenvalues 10 -1 10 Frequency (Hz) 0 10 1 Turbulent Simulation on Discrete Nodes (1) Simulated u-turbulence 10 node 6 node 1 10 0 -10 0 10 20 30 40 50 60 70 80 90 0 -10 100 0 -10 0 10 20 30 40 50 60 70 80 90 -10 100 node 3 node 8 0 10 20 30 40 50 60 70 80 90 100 node 9 node 4 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 10 0 0 10 20 30 40 50 60 70 80 90 0 -10 100 10 node 10 10 node 5 30 0 -10 10 0 -10 20 10 0 -10 10 0 10 -10 0 10 node 7 node 2 10 0 10 20 30 40 50 60 70 80 90 100 0 -10 Time (sec.) Time (sec.) Simulated w-turbulence 5 node 6 node 1 5 0 -5 0 10 20 30 40 50 60 70 80 90 100 node 7 node 2 0 0 10 20 30 40 50 60 70 80 90 node 3 node 8 0 10 20 30 40 50 60 70 80 90 100 node 9 node 4 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 5 0 0 10 20 30 40 50 60 70 80 90 0 -5 100 5 node 10 5 node 5 30 0 -5 5 0 -5 20 5 0 -5 10 0 -5 100 5 -5 0 5 5 -5 0 -5 0 10 20 30 40 50 Time (sec.) 60 70 80 90 100 0 -5 Time (sec.) Fig. 18 Simulated turbulent time series in some deck nodes at U=20m/s Turbulent Simulation on Discrete Nodes (2) Simulated u-turbulence 20 node 6 node 1 20 0 -20 0 10 20 30 40 50 60 70 80 90 100 0 0 10 20 30 40 50 60 70 80 90 100 node 3 node 8 0 10 20 30 40 50 60 70 80 90 100 node 9 node 4 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 20 0 0 10 20 30 40 50 60 70 80 90 0 -20 100 20 node 10 20 node 5 30 0 -20 20 0 -20 20 20 0 -20 10 0 -20 20 -20 0 20 node 7 node 2 20 -20 0 -20 0 10 20 30 40 50 60 70 80 90 0 -20 100 Time (sec.) Time (sec.) Simulated w-turbulence 10 node 6 node 1 10 0 -10 0 10 20 30 40 50 60 70 80 90 -10 100 0 0 10 20 30 40 50 60 70 80 90 100 node 3 node 8 0 10 20 30 40 50 60 70 80 90 100 node 9 node 4 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 10 0 0 10 20 30 40 50 60 70 80 90 0 -10 100 10 node 10 10 node 5 30 0 -10 10 0 -10 20 10 0 -10 10 0 -10 10 -10 0 10 node 7 node 2 10 -10 0 0 10 20 30 40 50 Time (sec.) 60 70 80 90 100 0 -10 Time (sec.) Fig. 19 Simulated turbulent time series in some deck nodes at U=30m/s Validation of Simulated Time Series u-turbulence 10 Suu(n) (m2/s) 10 10 10 10 10 10 3 10 1 2 10 1 Sww(n) (m2/s) 10 w-turbulence 0 -1 simulated at node 1 simulated at node 3 simulated at node 5 simulated at node 10 simulated at node 15 targeted spectrum -2 -3 10 -4 10 10 10 -2 10 -1 10 Frequency n(Hz) 0 10 1 0 -1 -2 simulated at node 1 simulated at node 3 simulated at node 5 simulated at node 10 simulated at node 15 targeted spectrum -3 10 -2 10 -1 10 0 Frequency n(Hz) Fig. 20 Verification on PSD of simulated turbulent time series at some nodes at U=20m/s 10 1 Effects of Numbers of Spectral Modes (1) u-turbulence w-turbulence Node 5: u(t) Node 5: w(t) 0 10 20 30 40 50 60 70 80 90 100 10 modes -10 0 10 0 10 20 30 40 50 60 70 80 90 100 20 modes -10 0 10 0 -10 0 10 10 20 30 40 50 60 70 80 90 100 0 -10 0 10 20 30 40 50 60 Time (sec.) 70 80 90 0 -5 0 5 20 30 40 50 60 70 80 90 100 10 20 30 40 50 60 70 80 90 100 0 -5 0 5 0 -5 0 5 20 40 10 20 u-turbulence 30 5 modes 30 40 50 60 70 80 90 100 10 modes 20 20 30 40 50 60 70 80 90 100 20 modes 10 0 10 20 30 40 50 60 70 80 90 100 30 modes 5 modes 10 modes 20 modes 30 modes Node 15 10 0 0 -10 0 40 50 60 Time (sec.) 70 80 90 100 5 0 -10 0 10 100 Node 15: w(t) 10 -10 0 10 80 w-turbulence Node 15: u(t) -10 0 10 60 0 -5 0 100 10 10 20 30 40 50 60 Time (sec.) 70 80 90 100 0 -5 0 5 10 20 30 40 50 60 70 80 90 100 10 20 30 40 50 60 70 80 90 100 10 20 30 40 50 60 70 80 90 100 10 20 30 40 50 60 70 80 90 100 0 -5 0 5 0 -5 0 5 0 -5 0 Time (sec.) Fig. 21 Effect of spectral modes on simulated time series Targeted time series 5 modes 5 30 modes 10 modes 20 modes 30 modes Node 5 5 modes 10 Effects of Numbers of Spectral Modes (2) u-turbulence 10 10 w-turbulence 1 10 1 30 modes 20 modes 10 modes 5 modes 0 30 modes 20 modes 10 modes 5 modes 10 0 -1 PSD 10 PSD Node 5 w-turbulence 10 10 10 -1 -2 10 -3 -4 10 -1 10 -2 -3 10 0 10 10 -1 10 1 Frequency (Hz) u-turbulence 10 10 1 10 1 1 30 modes 20 modes 10 modes 5 modes 0 10 -1 10 -2 10 -3 10 -1 10 10 w-turbulence PSD PSD Node 15 10 0 w-turbulence 30 modes 20 modes 10 modes 5 modes 10 10 Frequency (Hz) 0 -1 -2 -3 10 0 Frequency (Hz) 10 1 10 -1 10 10 0 Frequency (Hz) Fig. 21 Effect of spectral modes on PSD of simulated time series 10 1 Remarks and Insights Effect of number of the spectral turbulent modes on simulated time series has been investigated with verification for accuracy and consistence. It can be argued that it is not accurate enough for the turbulent simulation with using just few fundamental turbulent but many turbulent modes should be required modes, Thus, Digital Simulation of Turbulent Wind Fields basing on the meaning Covariance Matrix-based POD as well as modes Physical of the spectral eigenvalues and turbulent relating Better to Understandings Wind Fields basing on hidden events in on theTurbulent ongoing turbulent flow has been tried to establish. It is expected thatand theSpectral spectral eigenvalues can POD both Covariance MatrixMatrix-based characterize forwind scale measurements of the turbulent eddies of new the ongoing turbulent Analyses on will be challenging flow. However, further studies on the relationship between the spectral eigenvalues, turbulent modes and physical phenomena inside the turbulent structures will be needed for better undestandings.