Experiments on the stability of convection rolls in fluids whose viscosity depends on temperature

1978 ◽  
Vol 89 (3) ◽  
pp. 553-560 ◽  
Author(s):  
Frank M. Richter
Keyword(s):  
Rayleigh Number ◽  
Thermal Boundary Layer ◽  
Initial Conditions ◽  
Three Dimensional ◽  
Hexagonal Cell ◽  
Low Viscosity ◽  
Viscosity Increase ◽  
The Stability ◽  
Convection Rolls ◽  
Rayleigh Numbers

The stability of two-dimensional convection rolls has been studied as a function of the Rayleigh number, wavenumber and variation in viscosity. The experiments used controlled initial conditions for the wavenumber, Rayleigh numbers up to 25 000 and variations in viscosity up to a factor of about 20. The parameter range of stable rolls is bounded by a hexagonal-cell regime at small Rayleigh numbers and large variations in viscosity. Otherwise, the rolls are subject to the same transitions as have already been studied in fluids of uniform viscosity. The bimodal instability leading to a stable three-dimensional pattern occurs at smaller values of the average Rayleigh number as the variations in viscosity increase. This appears to be a consequence of the low viscosity of the warm thermal boundary layer associated with the original rolls.

Convective motions in a spherical shell

1985 ◽  
Vol 152 ◽  
pp. 39-48 ◽  
Author(s):  
Abdelfattah Zebib ◽  
Atul K. Goyal ◽  
Gerald Schubert
Keyword(s):  
Rayleigh Number ◽  
Spherical Shell ◽  
Spectral Method ◽  
Three Dimensional ◽  
Outer Radius ◽  
Nonlinear Convection ◽  
Convective Motions ◽  
The Stability ◽  
Mode Truncation ◽  
Rayleigh Numbers

We compute the axisymmetric convective motions that exist in a spherical shell heated from below with inner to outer radius ratio equal to 0.5. The boundaries are stress-free and gravity is directly proportional to radius. Accurate solutions at large Rayleigh numbers (O(105)) are made feasible by a spectral method that employs diagonal-mode truncation. By examining the stability of axisymmetric motions we conclude that the preferred form of convection varies dramatically according to the value of the Rayleigh number. While axisymmetric motions with different patterns may exist for modestly nonlinear convection, only a single motion persists at sufficiently large values of the Rayleigh number. This circulation is symmetric about the equator and has two meridional cells with rising motion at the poles. Instability of this single axisymmetric motion determines that the preferred pattern of three-dimensional convection has one azimuthal wave.

Instabilities of convection rolls in a high Prandtl number fluid

1971 ◽  
Vol 47 (2) ◽  
pp. 305-320 ◽  
Author(s):  
F. H. Busse ◽  
J. A. Whitehead
Keyword(s):  
Prandtl Number ◽  
Three Dimensional ◽  
Wave Number ◽  
Stationary Convection ◽  
Wave Numbers ◽  
Theoretical Predictions ◽  
High Prandtl Number ◽  
The Stability ◽  
Convection Rolls ◽  
Rayleigh Numbers

An experiment on the stability of convection rolls with varying wave-number is described in extension of the earlier work by Chen & Whitehead (1968). The results agree with the theoretical predictions by Busse (1967a) and show two distinct types of instability in the form of non-oscillatory disturbances. The ‘zigzag instability’ corresponds to a bending of the original rolls; in the ‘cross-roll instability’ rolls emerge at right angles to the original rolls. At Rayleigh numbers above 23,000 rolls are unstable for all wave-numbers and are replaced by a three-dimensional form of stationary convection for which the name ‘bimodal convection’ is proposed.

scholarly journals A Fractional-Order Discrete Noninvertible Map of Cubic Type: Dynamics, Control, and Synchronization

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-21
Author(s):  
Xiaojun Liu ◽  
Ling Hong ◽  
Lixin Yang ◽  
Dafeng Tang
Keyword(s):  
Fractional Order ◽  
Initial Conditions ◽  
Dimensional Space ◽  
Equilibrium Points ◽  
Three Dimensional ◽  
State Variables ◽  
Discrete Maps ◽  
The Stability ◽  
Control And Synchronization ◽  
Simultaneous Variation

In this paper, a new fractional-order discrete noninvertible map of cubic type is presented. Firstly, the stability of the equilibrium points for the map is examined. Secondly, the dynamics of the map with two different initial conditions is studied by numerical simulation when a parameter or a derivative order is varied. A series of attractors are displayed in various forms of periodic and chaotic ones. Furthermore, bifurcations with the simultaneous variation of both a parameter and the order are also analyzed in the three-dimensional space. Interior crises are found in the map as a parameter or an order varies. Thirdly, based on the stability theory of fractional-order discrete maps, a stabilization controller is proposed to control the chaos of the map and the asymptotic convergence of the state variables is determined. Finally, the synchronization between the proposed map and a fractional-order discrete Loren map is investigated. Numerical simulations are used to verify the effectiveness of the designed synchronization controllers.

The onset of convective instability in a triply diffusive fluid layer

1989 ◽  
Vol 202 ◽  
pp. 443-465 ◽  
Author(s):  
Arne J. Pearlstein ◽  
Rodney M. Harris ◽  
Guillermo Terrones
Keyword(s):  
Rayleigh Number ◽  
Linear Stability ◽  
Fluid Layer ◽  
Neutral Curve ◽  
Interesting Consequence ◽  
Stability Criteria ◽  
Neutral Curves ◽  
The Stability ◽  
Rayleigh Numbers ◽  
Periodic Bifurcation

The onset of instability is investigated in a triply diffusive fluid layer in which the density depends on three stratifying agencies having different diffusivities. It is found that, in some cases, three critical values of the Rayleigh number are required to specify the linear stability criteria. As in the case of another problem requiring three Rayleigh numbers for the specification of linear stability criteria (the rotating doubly diffusive case studied by Pearlstein 1981), the cause is traceable to the existence of disconnected oscillatory neutral curves. The multivalued nature of the stability boundaries is considerably more interesting and complicated than in the previous case, however, owing to the existence of heart-shaped oscillatory neutral curves. An interesting consequence of the heart shape is the possibility of ‘quasi-periodic bifurcation’ to convection from the motionless state when the twin maxima of the heart-shaped oscillatory neutral curve lie below the minimum of the stationary neutral curve. In this case, there are two distinct disturbances, with (generally) incommensurable values of the frequency and wavenumber, that simultaneously become unstable at the same Rayleigh number. This work complements the earlier efforts of Griffiths (1979a), who found none of the interesting results obtained herein.

Convective Instability in a Melt Layer Heated From Below

1976 ◽  
Vol 98 (1) ◽  
pp. 88-94 ◽  
Author(s):  
E. M. Sparrow ◽  
L. Lee ◽  
N. Shamsundar
Keyword(s):  
Rayleigh Number ◽  
Phase Change ◽  
Wall Temperature ◽  
A Priori ◽  
Linear Stability Theory ◽  
Convective Heating ◽  
Fluid Medium ◽  
Lower Bounding ◽  
The Stability ◽  
Rayleigh Numbers

Consideration is given to the onset of convective motions in a horizontal melt layer created by solid-to-liquid phase change. The melt layer is heated at its lower bounding surface either due to convective transfer from an adjacent fluid medium or to a step change in wall temperature. The analysis is carried out for liquid melts whose densities decrease with increasing temperature. Linear stability theory is employed to determine the conditions marking the onset of motion. The results of the analysis are expressed in terms of two Rayleigh numbers. One of these, the internal Rayleigh number, is based on the instantaneous thickness and instantaneous temperature difference across the layer. The other, the external Rayleigh number, is more convenient to use in applications problems since it contains quantities which are constant and a priori prescribable. For a melting problem where the external Rayleigh number is large, instability occurs soon after the start of heating. At smaller external Rayleigh numbers, the duration time of the regime of no motion increases markedly. At large times, the stability results for convective heating coincide with those for stepped wall temperature. In addition to the results for the stability problem, results for conduction phase change (in the absence of motion) are also presented for the surface convection boundary condition.

scholarly journals The global characteristics of the three-dimensional thermal convection inside a spherical shell

1997 ◽  
Vol 4 (1) ◽  
pp. 19-27 ◽  
Author(s):  
J. Arkani-Hamed
Keyword(s):  
Boundary Layer ◽  
Nusselt Number ◽  
Rayleigh Number ◽  
Spherical Shell ◽  
Critical Rayleigh Number ◽  
Three Dimensional ◽  
Horizontal Layer ◽  
Boundary Layer Theory ◽  
Physical Parameters ◽  
Rayleigh Numbers

Abstract. The Rayleigh number-Nusselt number, and the Rayleigh number-thermal boundary layer thickness relationships are determined for the three-dimensional convection in a spherical shell of constant physical parameters. Several models are considered with Rayleigh numbers ranging from 1.1 x 102 to 2.1 x 105 times the critical Rayleigh number. At lower Rayleigh numbers the Nusselt number of the three-dimensional convection is greater than that predicted from the boundary layer theory of a horizontal layer but agrees well with the results of an axisymmetric convection in a spherical shell. At high Rayleigh numbers of about 105 times the critical value, which are the characteristics of the mantle convection in terrestrial planets, the Nusselt number of the three-dimensional convection is in good agreement with that of the boundary layer theory. At even higher Rayleigh numbers, the Nusselt number of the three-dimensional convection becomes less than those obtained from the boundary layer theory. The thicknesses of the thermal boundary layers of the spherical shell are not identical, unlike those of the horizontal layer. The inner thermal boundary is thinner than the outer one, by about 30- 40%. Also, the temperature drop across the inner boundary layer is greater than that across the outer boundary layer.

scholarly journals Natural convection with mixed insulating and conducting boundary conditions: low- and high-Rayleigh-number regimes

2014 ◽  
Vol 742 ◽  
pp. 636-663 ◽  
Author(s):  
P. Ripesi ◽  
L. Biferale ◽  
M. Sbragaglia ◽  
A. Wirth
Keyword(s):  
Natural Convection ◽  
Rayleigh Number ◽  
Boundary Conditions ◽  
Two Dimensions ◽  
Turbulent Regime ◽  
Thermal Insulating ◽  
Pattern Length ◽  
The Stability ◽  
The Impact ◽  
Rayleigh Numbers

AbstractWe investigate the stability and dynamics of natural convection in two dimensions, subject to inhomogeneous boundary conditions. In particular, we consider a Rayleigh–Bénard (RB) cell, where the horizontal top boundary contains a periodic sequence of alternating thermal insulating and conducting patches, and we study the effects of the heterogeneous pattern on the global heat exchange, at both low and high Rayleigh numbers. At low Rayleigh numbers, we determine numerically the transition from a regime characterized by the presence of small convective cells localized at the inhomogeneous boundary to the onset of ‘bulk’ convective rolls spanning the entire domain. Such a transition is also controlled analytically in the limit when the boundary pattern length is small compared with the cell vertical size. At higher Rayleigh number, we use numerical simulations based on a lattice Boltzmann method to assess the impact of boundary inhomogeneities on the fully turbulent regime up to $\mathit{Ra} \sim 10^{10}$.

scholarly journals Data assimilation for plume models

2005 ◽  
Vol 12 (2) ◽  
pp. 257-267 ◽  
Author(s):  
C. A. Hier Majumder ◽  
E. Bélanger ◽  
S. DeRosier ◽  
D. A. Yuen ◽  
A. P. Vincent
Keyword(s):  
Rayleigh Number ◽  
Data Assimilation ◽  
Initial Conditions ◽  
Adjoint Equations ◽  
Inertial Effects ◽  
Point Heat Source ◽  
Prediction Time ◽  
Prandtl Numbers ◽  
Rayleigh Numbers

Abstract. We use a four-dimensional variational data assimilation (4D-VAR) algorithm to observe the growth of 2-D plumes from a point heat source. In order to test the predictability of the 4D-VAR technique for 2-D plumes, we perturb the initial conditions and compare the resulting predictions to the predictions given by a direct numerical simulation (DNS) without any 4D-VAR correction. We have studied plumes in fluids with Rayleigh numbers between 106 and 107 and Prandtl numbers between 0.7 and 70, and we find the quality of the prediction to have a definite dependence on both the Rayleigh and Prandtl numbers. As the Rayleigh number is increased, so is the quality of the prediction, due to an increase of the inertial effects in the adjoint equations for momentum and energy. The horizon predictability time, or how far into the future the 4D-VAR method can predict, decreases as Rayleigh number increases. The quality of the prediction is decreased as Prandtl number increases, however. Quality also decreases with increased prediction time.

Stability Control of a Three-Dimensional Passive Walker by Periodic Input Based on the Frequency Entrainment

2011 ◽  
Vol 23 (6) ◽  
pp. 1100-1107 ◽  
Author(s):  
Soichiro Suzuki ◽  
◽  
Masamichi Takada ◽  
Yuta Iwakura ◽  
Keyword(s):  
Initial Conditions ◽  
Three Dimensional ◽  
Stability Control ◽  
Human Gait ◽  
Mechanical Oscillator ◽  
Frequency Entrainment ◽  
Passive Walking ◽  
Periodic Input ◽  
The Stability ◽  
Human Gait Analysis

This study proposes a new control that stabilizes a three-dimensional (3D) passive walker without torque input at knees and ankles joints by using entrainment and a mechanical oscillator. It is difficult to stabilize a 3D biped passive walker in different environments because the range of initial conditions for stable walking is limited, so we designed a 3D biped passive walker as a passive walking platform by considering the results of human gait analysis to make the success of passive walking high. The stability of this platform was analytically determined by analyzing the frontal movement limit cycle. In the new control, the frontalmovement period is synchronized with the swing-leg period by a mechanical oscillator on the top of the walker. The mechanical oscillator controller generates a target path to synchronize oscillatormovement with swing-leg movement using frequency entrainment. The walker is stabilized when the frontal movement period was synchronized with the swing-leg period by periodic input generated by the mechanical oscillator. It was experimentally found consequently that the walker was stabilized on different slopes and flat floors.

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