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TECHNICAL PAPERS: Forced Convection

Transition in Homogeneously Heated Inclined Plane Parallel Shear Flows

[+] Author and Article Information
S. Generalis

School of Engineering and Applied Sciences, Division of Chemical Engineering and Applied Chemistry, Aston University, United Kingdom

M. Nagata

Department of Aeronautics and Astronautics, Graduate School of Engineering, Kyoto University, Japan

J. Heat Transfer 125(5), 795-803 (Sep 23, 2003) (9 pages) doi:10.1115/1.1599370 History: Received November 14, 2002; Revised May 16, 2003; Online September 23, 2003
Copyright © 2003 by ASME
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References

Figures

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The geometrical configuration exhibiting the basic symmetric velocity profile U*(z*) of the plane-parallel shear flow in an inclined fluid layer heated internally. The temperature T*(z*) is measured from the environment.
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(a) The linear neutral curves in the (α,Gr) plane and for various values of the Prandtl number as indicated; (b) the critical wavenumber αc as a function of the Prandtl number; and (c) the critical Grashof number Grc as a function of the Prandtl number. χ=90 deg.
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The critical Grashof number for the primary and the second closed connected neutral curves for transverse roll type perturbations. Pr=7, β=0, and χ≈4.95 deg
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The closed neutral curves for (a) χ≈5.121 deg, (b) χ≈5.085 deg, and (c) χ≈4.95 deg. Pr=7. ⊗ corresponds to the value χ≈5.1273 deg.
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The critical values of the Grashof number (right scale—continuous curve) and the wavenumber (left scale—dash-dotted curve) against the angle of inclination χ for transverse roll type perturbations (β=0). Pr=7. ⊗ corresponds to the value χ≈5.1273 deg.
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The phase velocity (continuous curve—left scale) and real part of the leading eigenvalue, σ1r, (dashed curve—right scale) as a function of the Grashof number for the stability curve of Fig. 5 with χ≈4.95 deg and α=1.257, β=0, Pr=7
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The phase velocity (continuous curve—left scale) and real part of the leading eigenvalue, σ1r, (dashed curve—right scale) as a function of the Grashof number for the second lower stability curve of Fig. 5 with χ≈4.95 deg and α=2.01, β=0, Pr=7
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The neutral curve after the smooth merger of the two neutral curves for χ≈0.9 deg and β=0 with Pr=7
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The critical Grashof number as a function of the angle of inclination for longitudinal roll type perturbations (α=0, continuous curve) and for transverse roll type perturbations (β=0, dash-dotted disconnected curves). Pr=7.
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The critical wavenumber βc as a function of the angle of inclination χ for longitudinal roll type perturbations (α=0). Pr=7.
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The stream function of (a) the velocity fluctuations ∂ϕ/∂x, (b) the disturbance ∂ϕ/∂x+∫−1zŬdz, (c) the total flow, ∂ϕ/∂x+∫−1zU⁁dz, (d) the total temperature, for the secondary state with α=1.21, Gr=23,000, Pr=7. Colder regions are represented by the lighter shade.
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The total (a) mean flow (U⁁) profile and (b) mean temperature (T⁁) profile for various Grashof numbers and for a fixed wavenumber α=1.21. In (a) and (b) the dash-dotted, dashed, dash-dot-dotted and long-dashed curves correspond to Gr=4300, 12,000, 23,000 and 33,000, respectively. The solid curve in both figures represents the basic flow and temperature distributions. Pr=7.
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The real part of the leading eigenvalue as function of d for α=1.227, Gr=2817, Pr=7 and for fixed values of the parameter b as indicated. Note that for α=1.227, linear stability analysis predicts a value of Grc=2816.8898.
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Instability boundaries of secondary TWs for Pr=7. (1): d=0.01, (2): d=0.02, (3): d=0.03, (4): d=0.04, (a): b=0.2, (b): b=0.15, (c): b=0.1. For (1)–(4) b=0 and σ1i≠0. For (a)–(c) d=0 and σ1i=0. The outer curve represents the neutral curve.
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Instability boundaries of secondary TWs for Pr=0.71. (1): d=0.01, (2): d=0.03, (3): d=0.04, (a): b=0.2, (b): b=0.15, (c): b=0.1. For (1)–(3) b=0 and σ1i≠0. For (a)–(c) d=0 and σ1i≠0. The outer curve represents the neutral curve.

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