On Reduced Order Modeling of Transient Flows Using the Proper Orthogonal Decomposition

2012 ◽  
Author(s):  
Fariduddin Behzad ◽  
Brian T. Helenbrook ◽  
Goodarz Ahmadi
Keyword(s):  
Finite Element ◽  
Proper Orthogonal Decomposition ◽  
Stokes Equations ◽  
Reduced Order Modeling ◽  
Orthogonal Decomposition ◽  
Basis Functions ◽  
Navier Stokes ◽  
Reduced Order ◽  
Split Method ◽  
Proper Orthogonal

Reduced-order modeling (ROM) of transient fluid flows using the proper orthogonal decomposition (POD) was studied. Particular attention was given to incompressible, unsteady flow over a two-dimensional NACA0015 airfoil in the laminar regime. When the airfoil sheds vortices, a transient blowing through a jet placed at the 10% chord location was imposed. POD modes were derived from the numerical solution of the flow obtained using an hp-finite element method. The ROM was obtained by a streamwise-upwind-Petrov-Galerkin (SUPG) projection of the incompressible Navier–Stokes equations onto the space spanned by the POD modes. The extraction of accurate POD-based reduced order model of this flow was explored using three different POD mode generation methods. The first approach was the split method, which superposes modes derived from simulations of the blowing jet with no flow and simulations of the baseline flow with no jet. The second method combined POD modes derived from simulations having both the jet and flow with modes obtained from simulation of only the flow. These modes were generated after the simulations reached the periodic state. The third and newly proposed approach was to generate a set of modes called “Generalized POD basis functions.” These modes were derived from simulations where the jet’s flow amplitude is varied slowly. For each method, the results were compared with detailed Finite Element solutions and the accuracy and efficiency of different methods were evaluated. The newly proposed “Generalized POD basis functions” approach predicted the transient response of the system most accurately.

scholarly journals Investigation of the Effects of Length to Depth Ratio on Open Supersonic Cavities Using CFD and Proper Orthogonal Decomposition

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Ibrahim Yilmaz ◽  
Ece Ayli ◽  
Selin Aradag
Keyword(s):  
Proper Orthogonal Decomposition ◽  
Stokes Equations ◽  
Reduced Order Modeling ◽  
Orthogonal Decomposition ◽  
Navier Stokes ◽  
Reduced Order ◽  
Modeling Approach ◽  
Depth Ratio ◽  
Minimum Number ◽  
Proper Orthogonal

Simulations of supersonic turbulent flow over an open rectangular cavity are performed to observe the effects of length to depth ratio (L/D) of the cavity on the flow structure. Two-dimensional compressible time-dependent Reynolds-averaged Navier-Stokes equations with k-ωturbulence model are solved. A reduced order modeling approach, Proper Orthogonal Decomposition (POD) method, is used to further analyze the flow. Results are obtained for cavities with severalL/Dratios at a Mach number of 1.5. Mostly, sound pressure levels (SPL) are used for comparison. After a reduced order modeling approach, the number of modes necessary to represent the systems is observed for each case. The necessary minimum number of modes to define the system increases as the flow becomes more complex with the increase in theL/Dratio. This study provides a basis for the control of flow over supersonic open cavities by providing a reduced order model for flow control, and it also gives an insight to cavity flow physics by comparing several simulation results with different length to depth ratios.

scholarly journals Local improvements to reduced-order models using sensitivity analysis of the proper orthogonal decomposition

2009 ◽  
Vol 629 ◽  
pp. 41-72 ◽  
Author(s):  
ALEXANDER HAY ◽  
JEFFREY T. BORGGAARD ◽  
DOMINIQUE PELLETIER
Keyword(s):  
Sensitivity Analysis ◽  
Proper Orthogonal Decomposition ◽  
Stokes Equations ◽  
Orthogonal Decomposition ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Reduced Order ◽  
Selection Step ◽  
Low Dimensional ◽  
Proper Orthogonal

The proper orthogonal decomposition (POD) is the prevailing method for basis generation in the model reduction of fluids. A serious limitation of this method, however, is that it is empirical. In other words, this basis accurately represents the flow data used to generate it, but may not be accurate when applied ‘off-design’. Thus, the reduced-order model may lose accuracy for flow parameters (e.g. Reynolds number, initial or boundary conditions and forcing parameters) different from those used to generate the POD basis and generally does. This paper investigates the use of sensitivity analysis in the basis selection step to partially address this limitation. We examine two strategies that use the sensitivity of the POD modes with respect to the problem parameters. Numerical experiments performed on the flow past a square cylinder over a range of Reynolds numbers demonstrate the effectiveness of these strategies. The newly derived bases allow for a more accurate representation of the flows when exploring the parameter space. Expanding the POD basis built at one state with its sensitivity leads to low-dimensional dynamical systems having attractors that approximate fairly well the attractor of the full-order Navier–Stokes equations for large parameter changes.

scholarly journals A New Reduced Stabilized Mixed Finite-Element Method Based on Proper Orthogonal Decomposition for the Transient Navier-Stokes Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-19
Author(s):  
Aiwen Wang ◽  
Jian Li ◽  
Zhenhua Di ◽  
Xiangjun Tian ◽  
Dongxiu Xie
Keyword(s):  
Finite Element ◽  
Proper Orthogonal Decomposition ◽  
Stokes Equations ◽  
Mixed Finite Element ◽  
Mixed Finite Element Method ◽  
Orthogonal Decomposition ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Proper Orthogonal ◽  
Stabilized Mixed Finite Element

A reduced stabilized mixed finite-element (RSMFE) formulation based on proper orthogonal decomposition (POD) for the transient Navier-Stokes equations is presented. An ensemble of snapshots is compiled from the transient solutions derived from a stabilized mixed finite-element (SMFE) method based on two local Gauss integrations for the two-dimensional transient Navier-Stokes equations by using the lowest equal-order pair of finite elements. Then, the optimal orthogonal bases are reconstructed by implementing POD techniques for the ensemble snapshots. Combining POD with the SMFE formulation, a new low-dimensional and highly accurate SMFE method for the transient Navier-Stokes equations is obtained. The RSMFE formulation could not only greatly reduce its degrees of freedom but also circumvent the constraint of inf-sup stability condition. Error estimates between the SMFE solutions and the RSMFE solutions are derived. Numerical tests confirm that the errors between the RSMFE solutions and the SMFE solutions are consistent with the the theoretical results. Conclusion can be drawn that RSMFE method is feasible and efficient for solving the transient Navier-Stokes equations.

scholarly journals A Proper-Orthogonal Decomposition Variational Multiscale Approximation Method for a Generalized Oseen Problem

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
John Paul Roop
Keyword(s):  
Proper Orthogonal Decomposition ◽  
Incompressible Flows ◽  
Reduced Order Modeling ◽  
Orthogonal Decomposition ◽  
Qualitative Properties ◽  
Complex Flow ◽  
Reduced Order ◽  
Variational Multiscale ◽  
Oseen Problem ◽  
Proper Orthogonal

We introduce the variational multiscale (VMS) stabilization for the reduced-order modeling of incompressible flows. It is well known that the proper orthogonal decomposition (POD) technique in reduced-order modeling experiences numerical instability when applied to complex flow problems. In this case a POD discretization naturally separates out structures which corresponding to the energy cascade on large and small scales, in order, a VMS approach is natural. In this paper, we provide the mathematical background necessary for implementing VMS to a POD-Galerkin model of a generalized Oseen problem. We provide theoretical evidence which indicates the consistency of utilizing a VMS approach in the stabilization of reduced order flows. In addition we provide numerical experiments indicating that VMS improves fidelity in reproducing the qualitative properties of the flow.

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