Onset of natural convection in a differentially heated layer of gray and non-gray gas mixtures

Abstract

The onset of convection instability in a differentially heated layer consisting of gray and non-gray gaseous mixtures is studied numerically. The conditions investigated cover a wide range of Planck number values (\(Pl = \dfrac{\kappa k_T}{4\sigma T_0^3}\)), from the conduction-dominated regime of \(Pl\gg 1\) to the radiation-dominated regime of \(Pl \ll 1\). The linear stability theory is applied to mass, momentum and energy balance equations and the resulting linear stability equations are solved by Chebyshev spectral collocation method. The divergence of radiative flux is solved by the finite-volume-based discrete ordinates method. The Spectral Line Weighted sum of gray gas (SLW) model is used to represent the fine spectral variation of absorption coefficient for a non-gray gas medium. The results indicate that the critical Rayleigh number (\(Ra_c\)) for the onset of convection increases with mean temperature (\(T_0\)) in the conduction-dominated regime at low values of \(T_0\). In the radiation-dominated regime (\(Pl\ll 1\)), \(Ra_c\) decreases with \(T_0\) for gray media. If the medium is non-gray, the critical \(Ra_c\) reduces to even lower values (as compared with those of gray gases) due to the dependence of gas absorptivity on temperature \(T_0\). A reduction in the wall emissivity value increases the stability of the fluid layer due to reflection of radiation from the wall, in the radiation-dominated regime. The reverse trend is seen for \(Pl\gg 1\). The critical parameters also significantly depend on the concentrations of radiatively participating gases in the mixture. The temperature profile in the fluid layer transforms from a linear profile in conduction regime to a stratified profile with steep gradients near the walls, in the presence of non-gray participating gases.

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Abbreviations

\(C_p\) :

specific heat at constant pressure, J/kg K

d :

separation distance, m

G :

total irradiation, W/m\(^2\)

g :

acceleration due to gravity, m/s\(^2\)

I :

total intensity of radiation, W/m\(^2\) sr

\(k_T\) :

thermal conductivity, W/m K

Pl :

Planck number, \(\dfrac{\kappa k_T}{4\sigma T_0^3}\)

Pr :

Prandtl number, \(\dfrac{\nu }{\alpha }\)

q :

heat flux, W/m\(^2\)

Ra :

Rayleigh number, \(\dfrac{g\beta \Delta T d^3}{\nu \alpha }\)

T :

temperature, K

t :

time, s

\(T_1\), \(T_2\) :

bottom and top wall temperature respectively, K

\(T_0\) :

mean temperature, K

\(\vec {V}\) :

velocity vector, \(\vec {V}= u\vec {i} + v\vec {j} + w\vec {k}\)

W :

amplitude of velocity perturbation

Y :

species mole fraction

Z :

vertical distance, m

\(\alpha \) :

thermal diffusivity, m\(^2\)/s

\(\alpha _s\) :

non-dimensional static temperature gradient

\(\nu \) :

kinematic viscosity, kg/m-s

\(\rho \) :

density, kg/m\(^3\)

\(\kappa \) :

absorption coefficient, 1/m

\(\Omega \) :

solid angle, sr

\(\epsilon \) :

emissivity

\(\theta \) :

non-dimensional temperature field, \(\theta = \dfrac{(T-T_c)}{(T_h - T_c)}\)

\(\phi \) :

amplitude of temperature perturbation

\(\sigma \) :

Stefan–Boltzmann constant, \(5.67\times 10^{-8}\) W/m\(^2\) K\(^4\)

\(\tau \) :

optical thickness, \(\tau = \kappa L\)

b:

black body

j :

non-gray gas index

c:

conduction mode

r:

radiation mode

s:

static condition

w:

wall

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Correspondence to T Sundararajan.

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Cholake, S., Sundararajan, T. & Venkateshan, S.P. Onset of natural convection in a differentially heated layer of gray and non-gray gas mixtures. Sādhanā 46, 25 (2021). https://doi.org/10.1007/s12046-020-01542-8

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Keywords

  • Onset of convection
  • gray and non-gray gas mixtures
  • Planck number
  • stability