Weakly nonlinear analysis of Rayleigh–Bénard convection problem in extended Boussinesq approximation

2021 ◽  
Vol 408 ◽  
pp. 126374
Author(s):  
Mark Dostalík ◽  
Ctirad Matyska ◽  
Vít Průša
Keyword(s):  
Nonlinear Analysis ◽  
Boussinesq Approximation ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Analysis ◽  
Bénard Convection ◽  
Benard Convection ◽  
Convection Problem ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

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.

APPROXIMATION OF THE STATIONARY STATISTICAL PROPERTIES OF THE DYNAMICAL SYSTEM GENERATED BY THE TWO-DIMENSIONAL RAYLEIGH–BÉNARD CONVECTION PROBLEM

2011 ◽  
Vol 09 (04) ◽  
pp. 421-446 ◽  
Author(s):  
FLORENTINA TONE ◽  
XIAOMING WANG
Keyword(s):  
Dynamical System ◽  
Global Attractors ◽  
Statistical Properties ◽  
Two Dimensional ◽  
Time Step ◽  
Bénard Convection ◽  
Benard Convection ◽  
Convection Problem ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

In this article, we consider a temporal linear semi-implicit approximation of the two-dimensional Rayleigh–Bénard convection problem. We prove that the stationary statistical properties as well as the global attractors of this linear semi-implicit scheme converge to those of the 2D Rayleigh–Bénard problem as the time step approaches zero.

scholarly journals Buoyancy statistics in moist turbulent Rayleigh–Bénard convection

2010 ◽  
Vol 648 ◽  
pp. 509-519 ◽  
Author(s):  
JÖRG SCHUMACHER ◽  
OLIVIER PAULUIS
Keyword(s):  
Piecewise Linear ◽  
Three Dimensional ◽  
Boussinesq Approximation ◽  
Phase Changes ◽  
Bénard Convection ◽  
Benard Convection ◽  
Convection Layer ◽  
The Impact ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

We study shallow moist Rayleigh–Bénard convection in the Boussinesq approximation in three-dimensional direct numerical simulations. The thermodynamics of phase changes is approximated by a piecewise linear equation of state close to the phase boundary. The impact of phase changes on the turbulent fluctuations and the transfer of buoyancy through the layer is discussed as a function of the Rayleigh number and the ability to form liquid water. The enhanced buoyancy flux due to phase changes is compared with dry convection reference cases and related to the cloud cover in the convection layer. This study indicates that the moist Rayleigh–Bénard problem offers a practical framework for the development and evaluation of parameterizations for atmospheric convection.

A Model of Rayleigh-Bénard Convection

2013 ◽  
Author(s):  
Gary A. Glatzmaier
Keyword(s):  
Gravity Waves ◽  
Thermal Convection ◽  
Fluid Flows ◽  
Boussinesq Approximation ◽  
Internal Gravity Waves ◽  
Bénard Convection ◽  
Benard Convection ◽  
Key Characteristics ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

This chapter presents a model of Rayleigh–Bénard convection. It first describes the fundamental dynamics expected in a fluid that is convectively stable and in one that is convectively unstable, focusing on thermal convection and internal gravity waves. Thermal convection and internal gravity waves are the two basic types of fluid flows within planets and stars that are driven by thermally produced buoyancy forces. The chapter then reviews the equations that govern fluid dynamics based on conservation of mass, momentum, and energy. It also examines the conditions under which the Boussinesq approximation simplifies conservation equations to a form very similar to that of an incompressible fluid. Finally, it discusses the key characteristics of the model of Rayleigh–Bénard convection.

Measuring the departures from the Boussinesq approximation in Rayleigh–Bénard convection experiments

2011 ◽  
Vol 682 ◽  
pp. 543-557 ◽  
Author(s):  
H. KURTULDU ◽  
K. MISCHAIKOW ◽  
M. F. SCHATZ
Keyword(s):  
Algebraic Topology ◽  
Betti Numbers ◽  
Boussinesq Approximation ◽  
Experimental Conditions ◽  
Bénard Convection ◽  
Benard Convection ◽  
The Difference ◽  
Convection Patterns ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

Algebraic topology (homology) is used to characterize quantitatively non-Oberbeck–Boussinesq (NOB) effects in chaotic Rayleigh–Bénard convection patterns from laboratory experiments. For fixed parameter values, homology analysis yields a set of Betti numbers that can be assigned to hot upflow and, separately, to cold downflow in a convection pattern. An analysis of data acquired under a range of experimental conditions where NOB effects are systematically varied indicates that the difference between time-averaged Betti numbers for hot and cold flows can be used as an order parameter to measure the strength of NOB-induced pattern asymmetries. This homology-based measure not only reveals NOB effects that Fourier methods and measurements of pattern curvature fail to detect, but also permits distinguishing pattern changes caused by modified lateral boundary conditions from NOB pattern changes. These results suggest a new approach to characterizing data from either experiments or simulations where NOB effects are expected to play an important role.

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