Asymptotic study and weakly nonlinear analysis at the onset of Rayleigh–Bénard convection in Hele–Shaw cell

1995 ◽  
Vol 7 (5) ◽  
pp. 926-934 ◽  
Author(s):  
Saïd Aniss ◽  
Mohamed Souhar ◽  
Jean Pierre Brancher
Keyword(s):  
Nonlinear Analysis ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Analysis ◽  
Asymptotic Study ◽  
Bénard Convection ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard ◽  
Hele Shaw Cell

Linear and weakly nonlinear analysis of Rayleigh–Bénard convection of perfect gas with non-Oberbeck–Boussinesq effects

2018 ◽  
Vol 845 ◽  
pp. 141-169 ◽  
Author(s):  
Shuang Liu ◽  
Shu-Ning Xia ◽  
Rui Yan ◽  
Zhen-Hua Wan ◽  
De-Jun Sun
Keyword(s):  
Nonlinear Analysis ◽  
Linear Stability ◽  
Flow Instabilities ◽  
Amplitude Equations ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Analysis ◽  
Bénard Convection ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

The influences of non-Oberbeck–Boussinesq (NOB) effects on flow instabilities and bifurcation characteristics of Rayleigh–Bénard convection are examined. The working fluid is air with reference Prandtl number $Pr=0.71$ and contained in two-dimensional rigid cavities of finite aspect ratios. The fluid flow is governed by the low-Mach-number equations, accounting for the NOB effects due to large temperature difference involving flow compressibility and variations of fluid viscosity and thermal conductivity with temperature. The intensity of NOB effects is measured by the dimensionless temperature differential $\unicode[STIX]{x1D716}$. Linear stability analysis of the thermal conduction state is performed. An $\unicode[STIX]{x1D716}^{2}$ scaling of the leading-order corrections of critical Rayleigh number $Ra_{cr}$ and disturbance growth rate $\unicode[STIX]{x1D70E}$ due to NOB effects is identified, which is a consequence of an intrinsic symmetry of the system. The influences of weak NOB effects on flow instabilities are further studied by perturbation expansion of linear stability equations with regard to $\unicode[STIX]{x1D716}$, and then the influence of aspect ratio $A$ is investigated in detail. NOB effects are found to enhance (weaken) flow stability in large (narrow) cavities. Detailed contributions of compressibility, viscosity and buoyancy actions on disturbance kinetic energy growth are identified quantitatively by energy analysis. Besides, a weakly nonlinear theory is developed based on centre-manifold reduction to investigate the NOB influences on bifurcation characteristics near convection onset, and amplitude equations are constructed for both codimension-one and -two cases. Rich bifurcation regimes are observed based on amplitude equations and also confirmed by direct numerical simulation. Weakly nonlinear analysis is useful for organizing and understanding these simulation results.

Strongly nonlinear Langmuir circulation and Rayleigh–Bénard convection

2008 ◽  
Vol 614 ◽  
pp. 39-65 ◽  
Author(s):  
G. P. CHINI
Keyword(s):  
Prandtl Number ◽  
Analytical Prediction ◽  
Langmuir Circulation ◽  
Asymptotic Results ◽  
Weakly Nonlinear ◽  
Strongly Nonlinear ◽  
Bénard Convection ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

Most rational asymptotic studies of non-rotating Rayleigh–Bénard convection and its cousins have been restricted to the linear or weakly nonlinear regime. An important exception occurs for large Rayleigh-number thermal convection at effectively infinite Prandtl number, i.e. fast but very viscous convection. In this scenario, the temperature field exhibits a layer-like structure surrounding an isothermal core and, crucially, the momentum equation linearizes. These features have been exploited by several authors to obtain semi-analytical nonlinear solutions. AtO(1) Prandtl number, the fluid dynamics in the vortex core is dominated by nonlinear inertial rather than linear viscous effects, substantially altering the vortex structure. Here, it is shown that a combination of matched asymptotic analysis and global conservation constraints can be used to obtain a semi-analytic yet strongly nonlinear description of two related flows: (i) Rayleigh–Bénard convection between constant heat-flux boundaries at unit Prandtl number, and (ii) Langmuir circulation (LC), a wind and wave-driven convective flow commonly observed in natural water bodies. A simple analytical prediction is given for the roll-vortex amplitude, which is shown to be independent of the horizontal wavenumber of the convection pattern. In marked contrast to weakly nonlinear convection cells, the fully nonlinear asymptotic solutions exhibit flow features relevant to turbulent convection including the complete vertical redistribution of the basic-state temperature (or, for LC, downwind velocity) field. Comparisons with well-resolved pseudospectral numerical simulations of the full two-dimensional governing equations confirm the accuracy of the asymptotic results.

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