A New Closure Strategy for Proper Orthogonal Decomposition Reduced-Order Models

Proper orthogonal decomposition (POD) is one of the most significant reduced-order modeling (ROM) techniques in fluid mechanics. However, the application of POD based reduced-order models (POD-ROMs) is primarily limited to laminar flows due to the decay of physical accuracy. A few nonlinear closure models have been developed for improving the accuracy and stability of the POD-ROMs, which are generally computationally expensive. In this paper we propose a new closure strategy for POD-ROMs that is both accurate and effective. In the new closure model, the Frobenius norm of the Jacobian of the POD-ROM is introduced as the eddy viscosity coefficient. As a first step, the new method has been tested on a one-dimensional Burgers equation with a small dissipation coefficient 1⁄4 10 . Numerical results show that the Jacobian based closure model greatly improves the physical accuracy of the POD-ROM, while maintaining a low computational cost. [DOI: 10.1115/1.4005928]

[1]  George Em Karniadakis,et al.  A low-dimensional model for simulating three-dimensional cylinder flow , 2002, Journal of Fluid Mechanics.

[2]  Arthur Veldman,et al.  Proper orthogonal decomposition and low-dimensional models for driven cavity flows , 1998 .

[3]  Ali H. Nayfeh,et al.  On the stability and extension of reduced-order Galerkin models in incompressible flows , 2009 .

[4]  K. Kunisch,et al.  Control of the Burgers Equation by a Reduced-Order Approach Using Proper Orthogonal Decomposition , 1999 .

[5]  Bérengère Podvin,et al.  On the adequacy of the ten-dimensional model for the wall layer , 2001 .

[6]  Nadine Aubry,et al.  The dynamics of coherent structures in the wall region of a turbulent boundary layer , 1988, Journal of Fluid Mechanics.

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

[8]  Anil E. Deane,et al.  Low-dimensional description of the dynamics in separated flow past thick airfoils , 1991 .

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

[10]  T. Iliescu,et al.  Modeling High Frequency Modes for Accurate Low-Dimensional Galerkin Models , 2009 .

[11]  Ali H. Nayfeh,et al.  Model Based Control of Laminar Wake Using Fluidic Actuation , 2010 .

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

[13]  Zhu Wang,et al.  Two-level discretizations of nonlinear closure models for proper orthogonal decomposition , 2011, J. Comput. Phys..

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

[15]  P. Sagaut.  Large Eddy Simulation for Incompressible Flows , 2001 .

[16]  John L. Lumley,et al.  Viscous Sublayer and Adjacent Wall Region in Turbulent Pipe Flow , 1967 .

[17]  Zhu Wang,et al.  Artificial viscosity proper orthogonal decomposition , 2011, Math. Comput. Model..

[18]  B. Podvin.  A proper-orthogonal-decomposition–based model for the wall layer of a turbulent channel flow , 2009 .

[19]  L. Berselli,et al.  Mathematics of Large Eddy Simulation of Turbulent Flows , 2005 .

[20]  L. Sirovich.  Turbulence and the dynamics of coherent structures. III. Dynamics and scaling , 1987 .

[21]  Hermann F. Fasel,et al.  Dynamics of three-dimensional coherent structures in a flat-plate boundary layer , 1994, Journal of Fluid Mechanics.

[22]  Nadine Aubry,et al.  Preserving Symmetries in the Proper Orthogonal Decomposition , 1993, SIAM J. Sci. Comput..

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

[24]  L Sirovich,et al.  Low-dimensional procedure for the characterization of human faces. , 1987, Journal of the Optical Society of America. A, Optics and image science.