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Research Papers: Natural and Mixed Convection

Effect of Phase Thermal Modulation Without Stationary Temperature Gradient on the Threshold of Convection

[+] Author and Article Information
K. Souhar

Laboratory of Energy Engineering,
Materials and Systems,
ENSA,
Ibn Zohr University,
Agadir 80000, Morocco
e-mail: k.souhar@uiz.ac.ma

S. Aniss

Laboratory of Mechanics,
Faculty of Sciences,
University Hassan II Casablanca,
20100, Morocco
e-mail: s.aniss@etude.univcasa.ma

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received December 24, 2015; final manuscript received May 12, 2016; published online June 7, 2016. Assoc. Editor: Andrey Kuznetsov.

J. Heat Transfer 138(10), 102502 (Jun 07, 2016) (8 pages) Paper No: HT-15-1814; doi: 10.1115/1.4033644 History: Received December 24, 2015; Revised May 12, 2016

The convective instability of a horizontal fluid layer subject to a time varying gradient of temperature is investigated. The stationary component of the temperature gradient is considered equal to zero and the oscillating components imposed on the horizontal boundaries are in phase and with the same amplitude. The aim of the present paper is to examine the effect of this type of modulation on the onset of convective instability. We show that unlike the case where the equilibrium configuration is stable in the absence of modulation, we have instability when the temperature at the horizontal boundaries is modulated in phase. Also, we observe that in the limit of low and high dimensionless frequency of modulation, ω < 0.5 and ω > 140, the basic state tends to a stable equilibrium configuration and for an intermediate dimensionless frequency, the system is potentially unstable. The results obtained from analytical asymptotic study for low and high dimensionless frequency are in good agreement with the numerical ones.

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Figures

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Fig. 1

Horizontal fluid layer of infinite extension in the x* and y* directions with a phase modulation of the temperature at the upper and lower surfaces

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Fig. 2

Time evolution of the equilibrium temperature profile with respect to the vertical dimensionless coordinate z, for different values of the dimensionless frequency, ω, over half a period 0≤t≤T/2=π/ω

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Fig. 3

Marginal stability curves for ω = 112.5 and Pr = 7. (a) Free–free boundaries and (b) rigid–rigid boundaries.

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Fig. 4

Numerical evolution and asymptotic behavior of the critical Rayleigh number, Rac, versus 1/ω for free–free and rigid–rigid boundaries.  (1)Rac=8501.77 σ−2, (2)Rac=18670.99 σ−2,  (3)Rac=37776.99 σ−4, (4)Rac=81542.86 σ−4.

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Fig. 5

Numerical evolution and asymptotic behavior of the critical wave number, qc, versus 1/ω for high frequencies. (a) Free–free boundaries, qc=0.373ω and (b) rigid–rigid boundaries, qc=0.44ω.

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Fig. 6

Numerical evolution and asymptotic behavior of the critical Rayleigh number, Rac, versus 1/ω in the limit of high frequencies in the both cases of boundaries, free–free and rigid–rigid

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