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Research Papers: Forced Convection

Surface Shape Design in Different Convection Heat Transfer Problems Via a Novel Coupled Algorithm

[+] Author and Article Information
Mehdi Nikfar

Young Researchers and Elite Club,
Islamic Azad University,
Lahijan Branch,
Eastern Kashef Street, P.O. Box: 1616,
Lahijan, Guilan, Iran
e-mail: nikfar.mehdi@gmail.com

Peyman Mayeli

Young Researchers and Elite Club,
Islamic Azad University,
Lahijan Branch,
Eastern Kashef Street, P.O. Box: 1616,
Lahijan, Guilan, Iran
e-mail: peyman.mayeli@gmail.com

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received February 6, 2017; final manuscript received July 3, 2017; published online September 19, 2017. Assoc. Editor: Milind A. Jog.

J. Heat Transfer 140(2), 021702 (Sep 19, 2017) (15 pages) Paper No: HT-17-1066; doi: 10.1115/1.4037581 History: Received February 06, 2017; Revised July 03, 2017

In this study, a new coupled surface shape design (SSD) methodology named direct design method is presented for the solution of problems containing different types of convection heat transfer in which a specific distribution of either heat flux or temperature is given instead of the shape of a boundary. In the proposed method, the governing equation, without using any mathematical transformation for the physical domains, is manipulated so that the grid generation, solving fluid flow, and heat transfer as well as shape updating can all be carried out simultaneously. Five different inverse shape design problems containing different types of convection heat transfer are solved by the proposed method. All the problems are also solved using the ball-spine algorithm (BSA), which is a recently developed de-coupled algorithm, for the sake of comparison. In all problems, the effects of using different under-relaxation parameters are investigated and the capability of both approaches is compared with each other. The results show that the proposed coupled method can solve the problems better than the BSA in the sense that the direct design method converges sooner than the BSA when the same under-relaxation parameter is used for both methods. Also, it is shown that the computational cost of solving a SSD problem using the direct design method is slightly greater than solving an analysis problem.

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Figures

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Fig. 1

Test cases of SSD problems with forced convection heat transfer: (a) cylindrical quote flow and (b) diffuser

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Fig. 2

Test cases of SSD problems with natural convection and conjugate heat transfer: (a) natural convection in a concentric annulus and (b) conjugate heat transfer test case

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Fig. 3

Test cases of SSD problems with mixed convection heat transfer

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Fig. 4

The computational grid used in this study

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Fig. 5

Local coordinates in a typical quadrilateral element

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Fig. 6

The flowchart of the direct design algorithm

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Fig. 7

Schematic of the ball spine algorithm used in 2D SSD problems: (a) internal flow problems and (b) external flow problems

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Fig. 8

The flowchart of the BSA

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Fig. 9

Heat flux distributions for initial and target geometries in the cylindrical Couette flow problem

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Fig. 10

Results of the first test case at Re = 100: (a) convergence histories, (b) shape evolutions during several iterations, and (c) a comparison between designed and target geometry

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Fig. 11

Results of the second test case at Re=50 and Pr=7.02 : (a) initial and target heat flux distributions, (b) initial and computed geometries corresponding to the defined heat flux distributions, (c) streamlines and isotherms in the designed geometry in case of S*=0.4, and (d) comparison of the computational cost between direct design approach and the BSA method with equal under-relation value (ω for direct design and C for BSA)

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Fig. 12

Results of the third test case at Ra=103 and Pr=0.71 : (a) initial and target heat flux distributions along the outer cylinder, (b) shape evolutions corresponding to the direct design with ω=0.5, (c) comparing the convergence rate of the direct design approach with the BSA method with equal under-relation value (ω for direct design and C for BSA), and (d) the contour of dimensionless temperature (right half) and streamlines (left half) in the designed geometry

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Fig. 13

Results of the forth test case at θs=0.85, kf/ks=0.1, Pr = 0.71: (a) computed shapes for different Reynolds numbers and (b) comparing the convergence rate of the direct design approach with the BSA method at Re = 20 with equal under-relation value (ω for direct design and C for BSA)

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Fig. 14

Results of the fifth test case at Ri = 1, Gr = 104 and Re = 100: (a) initial and target heat flux distributions along the bottom surface, (b) shape evolutions during several iterations corresponding to the direct design with ω=0.6, and (c) convergence rate of the direct design approach for various values of under-relaxation factor

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