POD-spectral decomposition for fluid flow analysis and model reduction

We propose an algorithm that combines proper orthogonal decomposition with a spectral method to analyze and extract reduced order models of flows from time data series of velocity fields. The flows considered in this study are assumed to be driven by non-linear dynamical systems exhibiting a complex behavior within quasiperiodic orbits in the phase space. The technique is appropriate to achieve efficient reduced order models even in complex cases for which the flow description requires a discretization with a fine spatial and temporal resolution. The proposed analysis enables to decompose complex flow dynamics into modes oscillating at a single frequency. These modes are associated with different energy levels and spatial structures. The approach is illustrated using time-resolved PIV data of a cylinder wake flow with associated Reynolds number equal to 3,900.

[1]  B. O. Koopman,et al.  Hamiltonian Systems and Transformation in Hilbert Space. , 1931, Proceedings of the National Academy of Sciences of the United States of America.

[2]  L. Sirovich.  Turbulence and the dynamics of coherent structures. II. Symmetries and transformations , 1987 .

[3]  L. Sirovich.  Turbulence and the dynamics of coherent structures. I. Coherent structures , 1987 .

[4]  Parviz Moin,et al.  Characteristic-eddy decomposition of turbulence in a channel , 1989, Journal of Fluid Mechanics.

[5]  A. N. Sharkovskiĭ.  Dynamic systems and turbulence , 1989 .

[6]  Yan Y. Kagan,et al.  Fractal dimension of brittle fracture , 1991 .

[7]  I. Kevrekidis,et al.  Low‐dimensional models for complex geometry flows: Application to grooved channels and circular cylinders , 1991 .

[8]  P. Holmes,et al.  The Proper Orthogonal Decomposition in the Analysis of Turbulent Flows , 1993 .

[9]  P. Holmes,et al.  Turbulence, Coherent Structures, Dynamical Systems and Symmetry , 1996 .

[10]  M. M. Zdravkovich,et al.  Flow Around Circular Cylinders Volume 1: Fundamentals , 1997 .

[11]  A. Emami-Naeini,et al.  Nonlinear model reduction for simulation and control of rapid thermal processing , 1997, Proceedings of the 1997 American Control Conference (Cat. No.97CH36041).

[12]  A. Prasad,et al.  The instability of the shear layer separating from a bluff body , 1997, Journal of Fluid Mechanics.

[13]  Jiu Ding,et al.  The point spectrum of Frobenius-Perron and Koopman operators , 1998 .

[14]  George Em Karniadakis,et al.  A Spectral Vanishing Viscosity Method for Large-Eddy Simulations , 2000 .

[15]  George Em Karniadakis,et al.  Dynamics and low-dimensionality of a turbulent near wake , 2000, Journal of Fluid Mechanics.

[16]  B. R. Noack,et al.  A hierarchy of low-dimensional models for the transient and post-transient cylinder wake , 2003, Journal of Fluid Mechanics.

[17]  I. Mezić.  Spectral Properties of Dynamical Systems, Model Reduction and Decompositions , 2005 .

[18]  G. Karniadakis,et al.  A combined direct numerical simulation–particle image velocimetry study of the turbulent near wake , 2006, Journal of Fluid Mechanics.

[19]  Ionel Michael Navon,et al.  A reduced‐order approach to four‐dimensional variational data assimilation using proper orthogonal decomposition (Int. J. Numer. Meth. Fluids 2007; 53(10):1571–1583) , 2007 .

[20]  Ionel M. Navon,et al.  A reduced‐order approach to four‐dimensional variational data assimilation using proper orthogonal decomposition , 2007 .

[21]  Virginia Kalb,et al.  An intrinsic stabilization scheme for proper orthogonal decomposition based low-dimensional models , 2007 .

[22]  Guillermo Artana,et al.  Variational assimilation of POD low-order dynamical systems , 2007 .

[23]  E. Lamballais,et al.  Experimental and numerical studies of the flow over a circular cylinder at Reynolds number 3900 , 2008 .

[24]  L. Cordier,et al.  Optimal control of the cylinder wake in the laminar regime by Trust-Region methods and POD Reduced-Order Models , 2008 .

[25]  C. C. Pain,et al.  Reduced‐order modelling of an adaptive mesh ocean model , 2009 .

[26]  I. Mezić,et al.  Spectral analysis of nonlinear flows , 2009, Journal of Fluid Mechanics.

[27]  D. Henningson,et al.  Seventh IUTAM Symposium on Laminar-Turbulent Transition , 2010 .

[28]  Dan S. Henningson,et al.  Reduced-order models for flow control: balanced models and Koopman modes , 2010 .

[29]  S. Bagheri.  Analysis and control of transitional shear flows using global modes , 2010 .

[30]  P. Schmid,et al.  Dynamic mode decomposition of numerical and experimental data , 2008, Journal of Fluid Mechanics.

[31]  P. Schmid,et al.  Applications of the dynamic mode decomposition , 2011 .

[32]  Denis Sipp,et al.  Model reduction for fluids using frequential snapshots , 2011 .

[33]  Guillermo Artana,et al.  Strong and weak constraint variational assimilations for reduced order fluid flow modeling , 2012, J. Comput. Phys..

[34]  Clarence W. Rowley,et al.  Variants of Dynamic Mode Decomposition: Boundary Condition, Koopman, and Fourier Analyses , 2012, J. Nonlinear Sci..

[35]  B. R. Noack.  Turbulence, Coherent Structures, Dynamical Systems and Symmetry , 2013 .