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Research Papers: Natural and Mixed Convection

A Fast and Efficient Method for Predicting Fluid Flow and Heat Transfer Problems

[+] Author and Article Information
Peng Ding, Xue-Hong Wu, Ya-Ling He

State Key Laboratory of Multiphase Flow and Heat Transfer, School of Energy and Power Engineering, Xi’an Jiaotong University, Shaanxi 710049, P.R. China

Wen-Quan Tao

State Key Laboratory of Multiphase Flow and Heat Transfer, School of Energy and Power Engineering, Xi’an Jiaotong University, Shaanxi 710049, P.R. Chinawqtao@mail.xjtu.edu.cn

J. Heat Transfer 130(3), 032502 (Mar 06, 2008) (17 pages) doi:10.1115/1.2804935 History: Received August 31, 2006; Revised March 30, 2007; Published March 06, 2008

A fast and efficient method based on the proper orthogonal decomposition (POD) technique for predicting fluid flow and heat transfer problems is proposed in this paper. POD is first applied to an ensemble of numerical simulation results at design parameters to obtain the empirical coefficients and eigenfunctions, and then the fluid and temperature fields in the range of design parameters are resolved by a linear combination of empirical coefficients and eigenfunctions. The empirical coefficients at off-design parameters are obtained by a cubic spline interpolation method for steady problems and a Galerkin projection method for transient problems. Finally, the efficiency and accuracy of the algorithm are examined by three examples. The POD based algorithm can predict both the velocity and temperature fields in the range of design parameters accurately at a price of a large number of precomputed cases (snapshots). It also brings significant computational time savings for the new cases within the parameter range presimulated compared with the finite volume method with SIMPLE -like algorithm.

Copyright © 2008 by American Society of Mechanical Engineers
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Figures

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Figure 1

Geometry of the cavity and boundary conditions, h∕w=2

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Figure 2

Snapshots of temperature fields. (a) corresponds to the average temperature profile; (b)–(f) correspond to the fluctuation temperature fields at Ra=5000, Ra=50,000, Ra=100,000, 150,000, and Ra=200,000, respectively.

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Figure 3

The most dominant eight eigenfunctions obtained from the temperature snapshots by the POD technique

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Figure 4

The most dominant eight eigenfunctions obtained from the velocity snapshots by the POD technique (in streamline format)

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Figure 5

The relative error between the numerical solutions and the POD reconstructions at design parameters. (a) corresponds to the relative error between the numerical temperature fields and the POD reconstruction fields of the 40 snapshots; (b) corresponds to relative error between the numerical velocity fields and the POD reconstruction fields of the 40 snapshots.

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Figure 6

Comparison between the POD and the FVM solutions for Ra numbers at off-design parameters, with M=6. (a) and (b) correspond to the streamline and isothermal at Ra=17,000; (c) and (d) correspond to the streamline and isothermal at Ra=85,700; (e) and (f) correspond to streamline and isothermal at Ra=168,800.

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Figure 7

Schematic view of driven cavity flow

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Figure 8

The first six dominant velocity eigenfunctions obtained from the velocity snapshots by the POD technique

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Figure 9

The relative error between the numerical solution and the POD reconstruction at design parameters

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Figure 10

Relative error versus the truncation degree M at off-design parameters

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Figure 11

Comparison between the POD and the FVM solutions for Re=5300 with M=6

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Figure 12

Various shape of function f(t) used to examine the performance of the POD based algorithm. (a), (b), and (c) correspond to case (a), case (b), and case (c), respectively.

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Figure 14

The temporal variation of relative error between the numerical solutions and the POD solutions for different time-dependent function f(t). (a), (b), and (c) correspond to case (a), case (b), and case (c), respectively.

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Figure 15

The temporal variation of temperature at two points for different time-dependent functions f(t). (a) corresponds to the case (b); (b) corresponds to case (c).

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