scholarly journals A POD-BASED REDUCED-ORDER STABILIZED CRANK–NICOLSON MFE FORMULATION FOR THE NON-STATIONARY PARABOLIZED NAVIER–STOKES EQUATIONS

2015 ◽  
Vol 20 (3) ◽  
pp. 346-368 ◽  
Author(s):  
Zhendong Luo
Keyword(s):  
Degrees Of Freedom ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Mixed Finite Element ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Reduced Order ◽  
Algorithm Implementation ◽  
Proper Orthogonal ◽  
Crank Nicolson

We firstly employ a proper orthogonal decomposition (POD) method, Crank–Nicolson (CN) technique, and two local Gaussian integrals to establish a PODbased reduced-order stabilized CN mixed finite element (SCNMFE) formulation with very few degrees of freedom for non-stationary parabolized Navier–Stokes equations. Then, the error estimates of the reduced-order SCNMFE solutions, which are acted as a suggestion for choosing number of POD basis and a criterion for updating POD basis, and the algorithm implementation for the POD-based reduced-order SCNMFE formulation are provided, respectively. Finally, some numerical experiments are presented to illustrate that the numerical results are consistent with theoretical conclusions. Moreover, it is shown that the reduced-order SCNMFE formulation is feasible and efficient for finding numerical solutions of the non-stationary parabolized Navier–Stokes equations.

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.

Influence of Wall Proximity on the Lift and Drag of a Particle in an Oscillatory Flow

2004 ◽  
Vol 127 (3) ◽  
pp. 583-594 ◽  
Author(s):  
Paul F. Fischer ◽  
Gary K. Leaf ◽  
Juan M. Restrepo
Keyword(s):  
Lift Force ◽  
Degrees Of Freedom ◽  
Oscillatory Flow ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Drag Forces ◽  
Lift And Drag ◽  
Two Parameters

We report on the lift and drag forces on a stationary sphere subjected to a wall-bounded oscillatory flow. We show how these forces depend on two parameters, namely, the distance between the particle and the bounding wall, and on the frequency of the oscillatory flow. The forces were obtained from numerical solutions of the unsteady incompressible Navier–Stokes equations. For the range of parameters considered, a spectral analysis found that the forces depended on a small number of degrees of freedom. The drag force manifested little change in character as the parameters varied. On the other hand, the lift force varied significantly: We found that the lift force can have a positive as well as a negative time-averaged value, with an intermediate range of external forcing periods in which enhanced positive lift is possible. Furthermore, we determined that this force exhibits a viscous-dominated and a pressure-dominated range of parameters.

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.

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