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Technical Briefs

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

[+] Author and Article Information
Imran Akhtar1

Department of Mechanical Engineering, NUST College of Electrical & Mechanical Engineering,  National University of Sciences & Technology (NUST), Islamabad, Pakistanimran.akhtar@ceme.nust.edu.pk

Zhu Wang, Jeff Borggaard, Traian Iliescu

Interdisciplinary Center for Applied Mathematics, MC 0531, Virginia Tech, Blacksburg, VA 24061

1

Corresponding author.

J. Comput. Nonlinear Dynam 7(3), 034503 (Apr 04, 2012) (6 pages) doi:10.1115/1.4005928 History: Received March 15, 2011; Revised December 19, 2011; Published April 03, 2012; Online April 04, 2012

## Abstract

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 $ν=10-3$. Numerical results show that the Jacobian based closure model greatly improves the physical accuracy of the POD-ROM, while maintaining a low computational cost.

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Copyright © 2012 by American Society of Mechanical Engineers
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## Figures

Figure 1

Schematic of energy distribution versus (a) wave number (LES approach) and (b) number of POD eigenfunctions (POD-ROM approach)

Figure 2

Numerical simulation of the Burgers equation with ν=10-3

Figure 3

Cumulative energy distribution of the POD modes for the Burgers equation with ν=10-3

Figure 4

POD-G ROM simulation of the Burgers equation with ν=10-3. Note that the results are inaccurate.

Figure 5

POD-L ROM simulation of the Burgers equation with ν=10-3. Note that the results have great improvement comparing with those of POD-G ROM.

Figure 6

POD-J ROM simulation of the Burgers equation with ν=10-3. Note that the results have great improvement comparing with those of POD-G ROM.

Figure 7

A comparison among DNS (blue), POD-G (green dash), POD-L (red triangle), and POD-J (cyan square) at t=1. Note that POD-J and POD-L are much closer to DNS than POD-G ROM. However, POD-J is computationally more efficient than POD-L.

Figure 8

Time evolution of POD basis coefficient qi, i=1, 2,…, 10 for DNS projection (blue), POD-G (green dash), POD-L (red triangle), and POD-J (cyan square). Note that the POD-L and POD-J models have close behaviors to DNS, and perform much better than the POD-G model.

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