Thermal Convection Induced by an Infinitesimally Thin and Unstably Stratified Layer

Abstract

We examine the linear stability analysis of the equations governing Rayleigh–Bénard convection flows when the basic temperature profile is unstably stratified solely over a thin horizontal slice of the fluid region. We conduct both asymptotic and numerical analyses on three distinct shapes of the basic temperature: (i) a hyperbolic tangent profile, (ii) a piecewise linear profile and (iii) a step-function profile. In the first two cases, the thin unstably stratified layer is centrally located. The presence of stably stratified regions below and above the central layer diminishes the effect of the velocity and thermal boundary layers that form at the plates. This in turn allows for the analysis of the convection process without the constraints of the horizontal boundaries to be simulated in a finite domain. We obtain expressions for the threshold parameters for convection onset as well as flow features as function of the thickness of the unstably stratified layer. In the limit of a vanishingly small thickness, the hyperbolic tangent profile tends to a step-function profile with a heavy top layer overlying a lighter bottom layer. These two layers are separated by an interface where a jump in density occurs. This situation resembles the Rayleigh–Taylor instability of a horizontal interface except that neither is the interface free nor is the buoyancy diffusion absent. The exploration of this case uncovers new instability threshold values and flow patterns. Finally, we discuss some relevant applications.

A step function density profile model for the convective stability of CO $$_2$$ 2 geological sequestration

“…Upon relaxing the function f (z) to be constant and applying the substitution, −p 2 = −ξ α 2 − Da, we solve Eqs. (9,10) in the two separate regions, 0 < z < Z 0 and Z 0 < z < 1 which we label with the super scripted variables W − (z),…”

“…The constants b and χ are measures of stable and unstable stratification, respectively, with the thickness of the unstably stratified layer getting thinner as χ increases resulting in a situation that mimics Bénard convection without the presence of the horizontal boundaries. Hadji et al [10] considered the case of a step change profile which would correspond to the SB profile in the limits of both b → 0 and χ → ∞, namely, T B (z) = T 1 + (T 2 − T 1 )H (z − Z 0 ), where T 1 and T 2 are the temperature values at the lower and upper plates, respectively, T 1 > T 2 and H (z) denotes the Heaviside function. Such a profile results in a heavy layer on top of a lighter one separated by a horizontal interface z = Z 0 across which diffusion of buoyancy takes place.…”

“…In the foregoing analysis, we consider the simple case of a reference state consisting of a unit step function that mimics the Rayleigh-Taylor instability. The resulting basic stratification is the most unstable and is expected to yield conservative values for the instability threshold conditions [10]. It offers, however, the advantage of conducting an analytically tractable investigation to explore the flow patterns through both linear and weakly nonlinear analyses.…”

A step function density profile model for the convective stability of CO2 geological sequestration

“…Upon relaxing the function f (z) to be constant and applying the substitution, −p 2 = −ξ α 2 − Da, we solve Eqs. (9,10) in the two separate regions, 0 < z < Z 0 and Z 0 < z < 1 which we label with the super scripted variables W − (z), W + (z), S − (z), and S + (z), respectively. With D = d/dz, we have…”

“…We derive an evolution equation for the leading order concentration profile by making use of long wavelength asymptotics valid in the limit β approaches zero, and as depicted in Fig. (10), the critical wavenumber approaches zero as β approaches zero. We invoke the analysis in [27] which has shown that the scaling of the critical wavenumber with the mass transfer coefficient β takes the form, α 4 C ∼ (β /h) as β → 0, for a dimensionless thickness of the plates of order unity.…”