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Research Papers: Natural and Mixed Convection

Scaling Analysis of the Unsteady Natural Convection Boundary Layer Adjacent to an Inclined Plate for Pr > 1 Following Instantaneous Heating

[+] Author and Article Information
Suvash C. Saha1

School of Engineering Systems,  Queensland University of Technology, Brisbane QLD 4001, Australiasuvash.saha@qut.edu.au

Feng Xu

Department of Mechanics, School of Civil Engineering,  Beijing Jiaotong University, Beijing 100044, China

Md Mamun Molla

Department of Mechanical Engineering and Manufacturing Engineering,  University of Manitoba, Winnipeg, R3T 5V6, Canada

1

Corresponding author.

J. Heat Transfer 133(11), 112501 (Aug 31, 2011) (9 pages) doi:10.1115/1.4004336 History: Received May 26, 2010; Revised May 26, 2011; Published August 31, 2011; Online August 31, 2011

The unsteady natural convection boundary layer adjacent to an instantaneously heated inclined plate is investigated using an improved scaling analysis and direct numerical simulations. The development of the unsteady natural convection boundary layer following instantaneous heating may be classified into three distinct stages including a start-up stage, a transitional stage, and a steady state stage, which can be clearly identified in the analytical and numerical results. Major scaling relations of the velocity and thicknesses and the flow development time of the natural convection boundary layer are obtained using triple-layer integral solutions and verified by direct numerical simulations over a wide range of flow parameters.

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Copyright © 2011 by American Society of Mechanical Engineers
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Figures

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Figure 1

Schematic of the computational domain and boundary conditions

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Figure 2

Snapshots of numerically simulated temperature contours (left) and streamlines (right) of the boundary-layer development at the start-up stage for Ra = 107 , Pr = 5 and A = 0.5, where τ is the nondimensional time defined in Sec. 4 (further refer to Eq. 31)

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Figure 3

A schematic of the temperature and velocity profiles normal to the inclined plate at its midpoint

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Figure 4

Time series of the maximum velocity along x = 0.5 with Ra = 1. 0 × 108 , Pr = 5 for A = 0.5

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Figure 5

Time series of the maximum velocity parallel to the plate at x = 0.5 for all cases considered: um (1 + A2 )1/2 (1 + Pr−1/2 )2 /A plotted against τ

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Figure 6

Velocity profiles parallel to the plate along the line x = 0.5 for at different times before steady state for all cases considered: (a) computed data and (b) u(1 + A2 )1/2 (1 + Pr−1/2 )2 /[] plotted against y(1 + Pr1/2 )Ra1/4 /τ1/2

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Figure 7

Temperature profiles along the line x = 0.5 for all cases considered at different times before steady state for all cases considered: (a) computed data and (b) θ plotted against yRa1/4 /τ1/2

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Figure 8

Velocity profiles parallel to the plate along the line x = 0.5 for all cases considered at steady state: (a) computed data and (b) u(1 + Pr−1/2 ) plotted against yRa1/4 A1/2 (1 + Pr−1/2 )1/2 /(1 + A2 )1/4

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Figure 9

Temperature profiles along the line x = 0.5 for all cases considered at steady state: (a) computed data and (b) θ plotted against yRa1/4 A1/2 /[(1 + A2 )1/4 (1 + Pr−1/2 )1/2 ]

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Figure 10

Time series of the maximum velocity parallel to the plate at x = 0.5 for all cases considered: (a) computed data and (b) um (1 + Pr−1/2 ) plotted against τA/[(1 + A2 )1/2 (1 + Pr−1/2 )]

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Figure 11

Nusselt number canculated on the heated plate for all cases considered: (a) computed data (b) Nu(1 + A2 )1/4 (1 + Pr−1/2 )1/2 /[A1/2 Ra1/4 ] plotted against τA/[(1 + A2 )1/2 (1 + Pr−1/2 )]

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