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TECHNICAL PAPERS

# Effects of Curvature and Convective Heat Transfer in Curved Square Duct Flows

[+] Author and Article Information
R. N. Mondal

Graduate School of Natural Science and Technology, Okayama University, Okayama 700-8530, Japan

Y. Kaga

Research Center for Energy Conversion, Doshisha University, Tataratsu-dani 1-3, Kyo-tanabe city, Kyoto 610-0394, Japan

T. Hyakutake

Department of Mechanical Engineering, Faculty of Engineering, Okayama University, Okayama 700-8530, Japan

S. Yanase

Department of Mechanical Engineering, Faculty of Engineering, Okayama University, Okayama 700-8530, Japanyanase@mech.okayama-u.ac.jp

J. Fluids Eng 128(5), 1013-1022 (Mar 03, 2006) (10 pages) doi:10.1115/1.2236131 History: Received June 16, 2005; Revised March 03, 2006

## Abstract

Non-isothermal flows with convective heat transfer through a curved duct of square cross section are numerically studied by using a spectral method, and covering a wide range of curvature, $δ$, $0<δ≤0.5$ and the Dean number, Dn, $0≤Dn≤6000$. A temperature difference is applied across the vertical sidewalls for the Grashof number $Gr=100$, where the outer wall is heated and the inner one cooled. Steady solutions are obtained by the Newton-Raphson iteration method and their linear stability is investigated. It is found that the stability characteristics drastically change due to an increase of curvature from $δ$ = 0.23 to 0.24. When there is no stable steady solution, time evolution calculations as well as their spectral analyses show that the steady flow turns into chaos through periodic or multi-periodic flows if Dn is increased no matter what $δ$ is. The transition to a periodic or chaotic state is retarded with an increase of $δ$. Nusselt numbers are calculated as an index of horizontal heat transfer and it is found that the convection due to the secondary flow, enhanced by the centrifugal force, increases heat transfer significantly from the heated wall to the fluid. If the flow becomes periodic and then chaotic, as Dn increases, the rate of heat transfer increases remarkably.

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## Figures

Figure 6

The results for δ=0.1 and Dn=3000. (a) Time evolution of λ and the values of λ for the steady solutions for 18≤t≤20. (b) Contours of secondary flow (top) and temperature profile (bottom) for 18.99≤t≤19.11.

Figure 3

Stability criterion on the Dn−δ plane (circle with broken line: boundary between the regions of stable and unstable solutions)

Figure 4

Contours of secondary flow (top) and temperature profile (bottom) on the first steady solution branch for various δ. (a)Dn=1000, (b)Dn=4000.

Figure 5

The results for δ=0.1 and Dn=1000. (a) Time evolution of λ and the values of λ for the steady solutions for 10≤t≤20. (b) Contours of secondary flow (top) and temperature profile (bottom) for 18.00≤t≤18.49.

Figure 7

The results for δ=0.1 and Dn=3500. (a) Time evolution of λ and the values of λ for the steady solutions for 22≤t≤30. (b) Contours of secondary flow (top) and temperature profile (bottom) for 25.95≤t≤27.0.

Figure 8

The results for δ=0.1 and Dn=3600. (a) Time evolution of λ and the values of λ for the steady solutions for 0≤t≤15. (b) Contours of secondary flow (top) and temperature profile (bottom) for 8.0≤t≤10.0.

Figure 9

The results for δ=0.1 and Dn=4000. (a) Time evolution of λ and the values of λ for the steady solutions for 0≤t≤10. (b) Contours of secondary flow (top) and temperature profile (bottom) for 6.0≤t≤8.0.

Figure 10

The results for δ=0.1 and Dn=5500. (a) Time evolution of λ and the values of λ for the steady solutions for 0≤t≤10. (b) Contours of secondary flow (top) and temperature profile (bottom) for 6.0≤t≤8.0.

Figure 11

Power spectra of the time evolution of λ at various Dn for δ=0.1. (a)Dn=3000, (b)Dn=3500, (c)Dn=3600, (d)Dn=4000, and (e)Dn=5500.

Figure 12

Distribution of the time-dependent solutions in the (Dn−δ) plane for 0≤Dn≤6000 and 0<δ≤0.5 (엯: steady-stable solution, ◻: periodic solution, ▵: chaotic solution, and the solid line shows a rough boundary of the periodic solutions)

Figure 13

Variation of the Nusselt number (Nu) with the Dean number (Dn) for the first steady solution branch at δ=0.1

Figure 14

Temperature gradients for δ=0.1(a) at the cooled sidewall and (b) at the heated sidewall

Figure 1

Coordinate system of the curved square duct

Figure 2

Steady solution branches for δ=0.1 and 100≤Dn≤2000 for Gr=100

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