Stability and resonant wave interactions of confined two-layer Rayleigh–Bénard systems

2014 ◽  
Vol 754 ◽  
pp. 415-455 ◽  
Author(s):  
S. V. Diwakar ◽  
Shaligram Tiwari ◽  
Sarit K. Das ◽  
T. Sundararajan
Keyword(s):  
Travelling Waves ◽  
Current Work ◽  
Wave Interactions ◽  
Resonant Wave ◽  
Aspect Ratios ◽  
Fluid Systems ◽  
Dimensional Parameters ◽  
The Individual ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

AbstractThe current work analyses the onset characteristics of Rayleigh–Bénard convection in confined two-dimensional two-layer systems. Owing to the interfacial interactions and the possibilities of convection onset in the individual layers, the two-layer systems typically exhibit diverse excitation modes. While the attributes of these modes range from the non-oscillatory mechanical/thermal couplings to the oscillatory standing/travelling waves, their regimes of occurrence are determined by the numerous system parameters and property ratios. In this regard, the current work aims at characterising their respective influence via methodical linear and fully nonlinear analyses, carried out on fluid systems that have been selected using the concept of balanced contrasts. Consequently, the occurrence of oscillatory modes is found to be associated with certain favourable fluid combinations and interfacial heights. The further branching of oscillatory modes into standing and travelling waves seems to additionally rely on the aspect ratio of the confined cavity. Specifically, the modulated travelling waves have been observed to occur (amidst standing wave modes) at discrete aspect ratios for which the onset of oscillatory convection happens at unequal fluid heights. This behaviour corresponds to the typical $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}m$:$n$ resonance where the critical wavenumbers of convection onset in the layers are dissimilar. Based on all of these observations, an attempt has been made in the present work to identify the oscillatory excitation modes with a reduced number of non-dimensional parameters.

Spatio-temporal regimes in Rayleigh–Bénard convection in a small rectangular cell

1989 ◽  
Vol 209 ◽  
pp. 309-334 ◽  
Author(s):  
M. A. Rubio ◽  
P. Bigazzi ◽  
L. Albavetti ◽  
S. Ciliberto
Keyword(s):  
Travelling Waves ◽  
Rectangular Cell ◽  
Complex Behaviour ◽  
Temporal Behaviour ◽  
Bénard Convection ◽  
Benard Convection ◽  
Spatio Temporal ◽  
Roll Axis ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

By means of an original optical technique we have studied the spatio-temporal behaviour in a Rayleigh–Bénard convection experiment of small rectangular geometry. The experimental technique allows complete reconstruction of the temperature field integrated along the roll axis. Two main spatiotemporal regimes have been found, corresponding to localized oscillations and travelling waves respectively. Several parameters are proposed for the quantitative characterization of this complex behaviour.

Has the ultimate state of turbulent thermal convection been observed?

2015 ◽  
Vol 785 ◽  
pp. 270-282 ◽  
Author(s):  
L. Skrbek ◽  
P. Urban
Keyword(s):  
Nusselt Number ◽  
Rayleigh Number ◽  
Ultimate State ◽  
Logarithmic Factor ◽  
Aspect Ratios ◽  
Bénard Convection ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Numbers ◽  
Rayleigh Benard

An important question in turbulent Rayleigh–Bénard convection is the scaling of the Nusselt number with the Rayleigh number in the so-called ultimate state, corresponding to asymptotically high Rayleigh numbers. A related but separate question is whether the measurements support the so-called Kraichnan law, according to which the Nusselt number varies as the square root of the Rayleigh number (modulo a logarithmic factor). Although there have been claims that the Kraichnan regime has been observed in laboratory experiments with low aspect ratios, the totality of existing experimental results presents a conflicting picture in the high-Rayleigh-number regime. We analyse the experimental data to show that the claims on the ultimate state leave open an important consideration relating to non-Oberbeck–Boussinesq effects. Thus, the nature of scaling in the ultimate state of Rayleigh–Bénard convection remains open.

Prograde, retrograde, and oscillatory modes in rotating Rayleigh–Bénard convection

2017 ◽  
Vol 831 ◽  
pp. 182-211 ◽  
Author(s):  
Susanne Horn ◽  
Peter J. Schmid
Keyword(s):  
Low Frequency ◽  
Linear Stability Theory ◽  
Dynamic Mode Decomposition ◽  
Dynamic Mode ◽  
Aspect Ratios ◽  
Bénard Convection ◽  
Mode Decomposition ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

Rotating Rayleigh–Bénard convection is typified by a variety of regimes with very distinct flow morphologies that originate from several instability mechanisms. Here we present results from direct numerical simulations of three representative set-ups: first, a fluid with Prandtl number $Pr=6.4$, corresponding to water, in a cylinder with a diameter-to-height aspect ratio of $\unicode[STIX]{x1D6E4}=2$; second, a fluid with $Pr=0.8$, corresponding to $\text{SF}_{6}$ or air, confined in a slender cylinder with $\unicode[STIX]{x1D6E4}=0.5$; and third, the main focus of this paper, a fluid with $Pr=0.025$, corresponding to a liquid metal, in a cylinder with $\unicode[STIX]{x1D6E4}=1.87$. The obtained flow fields are analysed using the sparsity-promoting variant of the dynamic mode decomposition (DMD). By means of this technique, we extract the coherent structures that govern the dynamics of the flow, as well as their associated frequencies. In addition, we follow the temporal evolution of single modes and present a criterion to identify their direction of travel, i.e. whether they are precessing prograde or retrograde. We show that for moderate $Pr$ a few dynamic modes suffice to accurately describe the flow. For large aspect ratios, these are wall-localised waves that travel retrograde along the periphery of the cylinder. Their DMD frequencies agree with the predictions of linear stability theory. With increasing Rayleigh number $Ra$, the interior gradually fills with columnar vortices, and eventually a regular pattern of convective Taylor columns prevails. For small aspect ratios and close enough to onset, the dominant flow structures are body modes that can precess either prograde or retrograde. For $Pr=0.8$, DMD additionally unveiled the existence of so far unobserved low-amplitude oscillatory modes. Furthermore, we elucidate the multi-modal character of oscillatory convection in low-$Pr$ fluids. Generally, more dynamic modes must be retained to accurately approximate the flow. Close to onset, the flow is purely oscillatory and the DMD reveals that these high-frequency modes are a superposition of oscillatory columns and cylinder-scale inertial waves. We find that there are coexisting prograde and retrograde modes, as well as quasi-axisymmetric torsional modes. For higher $Ra$, the flow also becomes unstable to wall modes. These low-frequency modes can both coexist with the oscillatory modes, and also couple to them. However, the typical flow feature of rotating convection at moderate $Pr$, the quasi-steady Taylor vortices, is entirely absent in low-$Pr$ flows.

scholarly journals Aspect ratio dependence of heat transport by turbulent Rayleigh–Bénard convection in rectangular cells

2012 ◽  
Vol 710 ◽  
pp. 260-276 ◽  
Author(s):  
Quan Zhou ◽  
Bo-Fang Liu ◽  
Chun-Mei Li ◽  
Bao-Chang Zhong
Keyword(s):  
Rayleigh Number ◽  
Heat Transport ◽  
Finite Conductivity ◽  
Turbulent Heat ◽  
Number Range ◽  
Aspect Ratios ◽  
Bénard Convection ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

AbstractWe report high-precision measurements of the Nusselt number $Nu$ as a function of the Rayleigh number $Ra$ in water-filled rectangular Rayleigh–Bénard convection cells. The horizontal length $L$ and width $W$ of the cells are 50.0 and 15.0 cm, respectively, and the heights $H= 49. 9$, 25.0, 12.5, 6.9, 3.5, and 2.4 cm, corresponding to the aspect ratios $({\Gamma }_{x} \equiv L/ H, {\Gamma }_{y} \equiv W/ H)= (1, 0. 3)$, $(2, 0. 6)$, $(4, 1. 2)$, $(7. 3, 2. 2)$, $(14. 3, 4. 3)$, and $(20. 8, 6. 3)$. The measurements were carried out over the Rayleigh number range $6\ensuremath{\times} 1{0}^{5} \lesssim Ra\lesssim 1{0}^{11} $ and the Prandtl number range $5. 2\lesssim Pr\lesssim 7$. Our results show that for rectangular geometry turbulent heat transport is independent of the cells’ aspect ratios and hence is insensitive to the nature and structures of the large-scale mean flows of the system. This is slightly different from the observations in cylindrical cells where $Nu$ is found to be in general a decreasing function of $\Gamma $, at least for $\Gamma = 1$ and larger. Such a difference is probably a manifestation of the finite plate conductivity effect. Corrections for the influence of the finite conductivity of the top and bottom plates are made to obtain the estimates of $N{u}_{\infty } $ for plates with perfect conductivity. The local scaling exponents ${\ensuremath{\beta} }_{l} $ of $N{u}_{\infty } \ensuremath{\sim} R{a}^{{\ensuremath{\beta} }_{l} } $ are calculated and found to increase from 0.243 at $Ra\simeq 9\ensuremath{\times} 1{0}^{5} $ to 0.327 at $Ra\simeq 4\ensuremath{\times} 1{0}^{10} $.

scholarly journals Lyapunov Exponents of Two Stochastic Lorenz 63 Systems

2019 ◽  
Vol 179 (5-6) ◽  
pp. 1343-1365 ◽  
Author(s):  
Bernard J. Geurts ◽  
Darryl D. Holm ◽  
Erwin Luesink
Keyword(s):  
Dynamical System ◽  
Lyapunov Exponents ◽  
Stochastic Version ◽  
System A ◽  
Fluctuation Dissipation ◽  
Total Phase ◽  
The Individual ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard ◽  
Space Volume

AbstractTwo different types of perturbations of the Lorenz 63 dynamical system for Rayleigh–Bénard convection by multiplicative noise—called stochastic advection by Lie transport (SALT) noise and fluctuation–dissipation (FD) noise—are found to produce qualitatively different effects, possibly because the total phase-space volume contraction rates are different. In the process of making this comparison between effects of SALT and FD noise on the Lorenz 63 system, a stochastic version of a robust deterministic numerical algorithm for obtaining the individual numerical Lyapunov exponents was developed. With this stochastic version of the algorithm, the value of the sum of the Lyapunov exponents for the FD noise was found to differ significantly from the value of the deterministic Lorenz 63 system, whereas the SALT noise preserves the Lorenz 63 value with high accuracy. The Lagrangian averaged version of the SALT equations (LA SALT) is found to yield a closed deterministic subsystem for the expected solutions which is isomorphic to the original Lorenz 63 dynamical system. The solutions of the closed chaotic subsystem, in turn, drive a linear stochastic system for the fluctuations of the LA SALT solutions around their expected values.

Scaling behaviour in Rayleigh–Bénard convection with and without rotation

2013 ◽  
Vol 717 ◽  
pp. 449-471 ◽  
Author(s):  
E. M. King ◽  
S. Stellmach ◽  
B. Buffett
Keyword(s):  
Thermal Boundary Layer ◽  
Scaling Behaviour ◽  
Bénard Convection ◽  
Fluid Systems ◽  
Horizontal Length ◽  
Benard Convection ◽  
Flow Speeds ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard ◽  
Quantities Of Interest

AbstractRotating Rayleigh–Bénard convection provides a simplified dynamical analogue for many planetary and stellar fluid systems. Here, we use numerical simulations of rotating Rayleigh–Bénard convection to investigate the scaling behaviour of five quantities over a range of Rayleigh ($1{0}^{3} \lesssim \mathit{Ra}\lesssim 1{0}^{9} $), Prandtl ($1\leq \mathit{Pr}\leq 100$) and Ekman ($1{0}^{- 6} \leq E\leq \infty $) numbers. The five quantities of interest are the viscous and thermal boundary layer thicknesses, ${\delta }_{v} $ and ${\delta }_{T} $, mean temperature gradients, $\beta $, characteristic horizontal length scales, $\ell $, and flow speeds, $\mathit{Pe}$. Three parameter regimes in which different scalings apply are quantified: non-rotating, weakly rotating and rotationally constrained. In the rotationally constrained regime, all five quantities are affected by rotation. In the weakly rotating regime, ${\delta }_{T} $, $\beta $ and $\mathit{Pe}$, roughly conform to their non-rotating behaviour, but ${\delta }_{v} $ and $\ell $ are still strongly affected by the Coriolis force. A summary of scaling results is given in table 2.

scholarly journals Aspect ratio effects in Rayleigh–Bénard convection of Herschel–Bulkley fluids

2017 ◽  
Vol 34 (5) ◽  
pp. 1658-1676 ◽  
Author(s):  
Mohammad Saeid Aghighi ◽  
Amine Ammar
Keyword(s):  
Heat Transfer ◽  
Finite Element ◽  
Rayleigh Number ◽  
Two Dimensional ◽  
Content Type ◽  
Bingham Number ◽  
Aspect Ratios ◽  
Power Law Index ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

Purpose The purpose of this paper is to analyze two-dimensional steady-state Rayleigh–Bénard convection within rectangular enclosures in different aspect ratios filled with yield stress fluids obeying the Herschel–Bulkley model. Design/methodology/approach In this study, a numerical method based on the finite element has been developed for analyzing two-dimensional natural convection of a Herschel–Bulkley fluid. The effects of Bingham number Bn and power law index n on heat and momentum transport have been investigated for a nominal Rayleigh number range (5 × 103 < Ra < 105), three different aspect ratios (ratio of enclosure length:height AR = 1, 2, 3) and a single representative value of nominal Prandtl number (Pr = 10). Findings Results show that the mean Nusselt number Nu¯ increases with increasing Rayleigh number due to strengthening of convective transport. However, with the same nominal value of Ra, the values of Nu¯ for shear thinning fluids n < 1 are greater than shear thickening fluids n > 1. The values of Nu¯ decrease with Bingham number and for large values of Bn, Nu¯ rapidly approaches unity, which indicates that heat transfer takes place principally by thermal conduction. The effects of aspect ratios have also been investigated and results show that Nu¯ increases with increasing AR due to stronger convection effects. Originality/value This paper presents a numerical study of Rayleigh–Bérnard flows involving Herschel–Bulkley fluids for a wide range of Rayleigh numbers, Bingham numbers and power law index based on finite element method. The effects of aspect ratio on flow and heat transfer of Herschel–Bulkley fluids are also studied.

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