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Control of Rayleigh-Bénard Convection in a Fluid Layer with Internal Heat Generation

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Abstract

The control of the onset of convection in a horizontal fluid layer with internal heat generation is studied. The horizontal boundaries of the system are cooled isothermally. The stability of the fluid layer is investigated on the basis of the linear stability theory and the resulting eigenvalues problem is solved numerically. Upon using a feedback proportional control, the heating power of the system is modulated in order to counteract any deviations of the temperature of the fluid from its conductive value. As a result, it is possible to postpone (or advance) significantly the onset of motion. The optimal positions of the thermal sensors can be predicted on the basis of the linear stability theory. The linear stability analysis also reveals the possible existence of Hopf’s bifurcations at the onset of motion. This type of bifurcation can be delayed using differential controllers. Two-dimensional numerical simulations of the full governing equations are carried out and found to agree well with the prediction of the linear stability theory.

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Abbreviations

A :

aspect ratio of the cavity

a :

wave number

C p :

specific heat at constant pressure of the fluid, J/(kgK)

D :

differential operator, d/dy

E′:

electric field magnitude, V/m

\(\vec {g}\) :

gravitational acceleration, m/s2

G′:

proportional controller’s gain, s− 1

G t :

differential controller’s gain

k :

thermal conductivity, W/(m.K)

L′:

height of fluid layer, m

P′:

pressure in the fluid layer, N/m2

Pr:

Prandtl number, ν/α

\(\dot {q}^{\prime }\) :

specific heating power, K/s

\(\dot {q}^{\prime }_{E} \) :

constant part of specific heating power, K/s

\(\dot {q}^{\prime }_{A} \) :

part of specific heating power supplied by the actuator, K/s

r :

ratio between the controlled and uncontrolled critical Rayleigh number

R a :

Rayleigh number, \(g\beta ^{\prime }\dot {q}^{\prime }_{E} L^{\prime 5} /\nu \alpha ^{2} \)

R a c :

critical Rayleigh number

t′:

time, s

T′:

temperature in the fluid layer, K

\(\vec {V}^{\prime }\) :

velocity vector in fluid layer, m/s

x′:

horizontal coordinate in fluid layer, m

y′:

vertical coordinate in fluid layer, m

\(y^{\prime }_{s} \) :

vertical position of the temperature sensor, m

α :

thermal diffusivity of the fluid, m2/s

β :

thermal expansion coefficient, K− 1

μ :

dynamic viscosity of fluid, Ns/m2

υ :

kinematic viscosity of fluid, m2/s

ρ :

density of fluid, kg/m3

σ :

electric conductivity, Ω− 1m− 1

c :

critical condition

r :

reference state

0:

value at y = 0

1:

value at y = 1

′:

refers to dimensional variable

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Authors and Affiliations

  1. Département de Génie Mécanique, Faculté de Technologie, Université Mustapha Ben Boulaid Batna 2, 05000, Batna, Algérie

    Z. Alloui

  2. Département de Génie Mécanique, Faculté des Sciences Appliquées, Université Kasdi Merbah Ouargla, 30000, Ouargla, Algérie

    Y. Alloui

  3. Ecole Polytechnique, Université de Montréal, C.P. 6079, Succ. Centre Ville, Montréal, QC, H3C 3A7, Canada

    P. Vasseur

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  1. Z. Alloui
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  2. Y. Alloui
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Correspondence to Z. Alloui.

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Alloui, Z., Alloui, Y. & Vasseur, P. Control of Rayleigh-Bénard Convection in a Fluid Layer with Internal Heat Generation. Microgravity Sci. Technol. 30, 885–897 (2018). https://doi.org/10.1007/s12217-018-9651-4

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