Numerical simulation of thermal convection in a two-dimensional finite box

1989 ◽  
Vol 199 ◽  
pp. 1-28 ◽  
Author(s):  
Isaac Goldhirsch ◽  
Richard B. Pelz ◽  
Steven A. Orszag
Keyword(s):  
Prandtl Number ◽  
Flow Patterns ◽  
Chaotic Regime ◽  
Two Dimensional ◽  
Onset Of Convection ◽  
Chaotic Flows ◽  
Chaotic Flow ◽  
Rayleigh Numbers ◽  
Pseudo Spectral ◽  
Rayleigh Benard

The problems of dynamical onset of convection, textural transitions and chaotic dynamics in a two-dimensional, rectangular Rayleigh-Bénard system have been investigated using well-resolved, pseudo-spectral simulations. All boundary conditions are taken to be no-slip. It is shown that the process of creating the temperature gradient in the system, is responsible for roll creation at the side boundaries. These rolls either induce new rolls or move into the interior of the cell, depending on the rate of heating. Complicated flow patterns and textural transitions are observed in both non-chaotic and chaotic flow regimes. Multistability is frequently observed. Intermediate-Prandtl-number fluids (e.g. 0.71) have a quasiperiodic time dependence up to Rayleigh numbers of order 106. When the Prandtl number is raised to 6.8, one observes aperiodic (chaotic) flows of non-integer dimension. In this case roll merging and separation is observed to be an important feature of the dynamics. In some cases corner rolls are observed to migrate into the interior of the cell and to grow into regular rolls; the large rolls may shrink and retreat into corners. The basic flow patterns observed do not change qualitatively when the chaotic regime is entered.

The effect of distant side-walls on the evolution and stability of finite-amplitude Rayleigh-Bénard convection

1981 ◽  
Vol 378 (1775) ◽  
pp. 539-566 ◽  
Keyword(s):  
Rayleigh Number ◽  
Prandtl Number ◽  
Finite Amplitude ◽  
Two Dimensional ◽  
Bénard Convection ◽  
Benard Convection ◽  
Side Walls ◽  
Rayleigh Benard Convection ◽  
Rayleigh Numbers ◽  
Rayleigh Benard

A recent study by Cross et al . (1980) has described a class of finite-amplitude phase-winding solutions of the problem of two-dimensional Rayleigh-Bénard convection in a shallow fluid layer of aspect ratio 2 L (≫ 1) confined laterally by rigid side-walls. These solutions arise at Rayleigh numbers R = R 0 + O ( L -1 ) where R 0 is the critical Rayleigh number for the corresponding infinite layer. Nonlinear solutions of constant phase exist for Rayleigh numbers R = R 0 + O ( L -2 ) but of these only the two that bifurcate at the lowest value of R are stable to two-dimensional linearized disturbances in this range (Daniels 1978). In the present paper one set of the class of phase-winding solutions is found to be stable to two-dimensional disturbances. For certain values of the Prandtl number of the fluid and for stress-free horizontal boundaries the results predict that to preserve stability there must be a continual readjustment of the roll pattern as the Rayleigh number is raised, with a corresponding increase in wavelength proportional to R - R 0 . These solutions also exhibit hysteresis as the Rayleigh number is raised and lowered. For other values of the Prandtl number the number of rolls remains unchanged as the Rayleigh number is raised, and the wavelength remains close to its critical value. It is proposed that the complete evolution of the flow pattern from a static state must take place on a number of different time scales of which t = O(( R - R 0 ) -1 ) and t = O(( R - R 0 ) -2 ) are the most significant. When t = O(( R - R 0 ) -1 ) the amplitude of convection rises from zero to its steady-state value, but the final lateral positioning of the rolls is only completed on the much longer time scale t = O(( R - R 0 ) -2 ).

An Analytical Study on Natural Convection in Isotropic and Anisotropic Porous Channels

1990 ◽  
Vol 112 (2) ◽  
pp. 396-401 ◽  
Author(s):  
T. Nilsen ◽  
L. Storesletten
Keyword(s):  
Analytical Study ◽  
Vertical Direction ◽  
Flow Patterns ◽  
Heat Conducting ◽  
Two Dimensional ◽  
Anisotropic Porous Media ◽  
Onset Of Convection ◽  
Linear Temperature ◽  
Weakly Nonlinear ◽  
Rayleigh Numbers

This paper is an analytical study on natural two-dimensional convection in horizontal rectangular channels filled by isotropic and anisotropic porous media. The channel walls, assumed to be impermeable and perfectly heat conducting, are nonuniformly heated to establish a linear temperature distribution in the vertical direction. We derive the critical Rayleigh numbers for the onset of convection and examine the steady flow patterns at moderately supercritical Rayleigh numbers. The stability properties of these flow patterns are examined against two-dimensional perturbations using a weakly nonlinear theory.

Randomly forced Rayleigh-Bénard convection

1980 ◽  
Vol 98 (2) ◽  
pp. 329-348 ◽  
Author(s):  
Bharat Jhaveri ◽  
G. M. Homsy
Keyword(s):  
Convective Transport ◽  
Time Intervals ◽  
Onset Of Convection ◽  
Nonlinear Convection ◽  
Bénard Convection ◽  
Benard Convection ◽  
Random Fluctuations ◽  
Rayleigh Benard Convection ◽  
Rayleigh Numbers ◽  
Rayleigh Benard

We consider the onset of Rayleigh–Bénard convection from random fluctuations arising within a fluid. In the specific case in which the fluctuations are thermodynamically determined, we reduce the problem to a random initial value problem for the Fourier modes. For the case of weak nonlinear convection, it is possible to truncate the number of modes and this truncated set is solved both by a Monte Carlo technique and by moment methods for various Rayleigh numbers. We find three stages in the evolution of ordered convection from random fluctuations which correspond to time intervals in which the fluctuations and the nonlinearity have different degrees of importance. It is shown that no simple moment truncation method will succeed and that the time for onset of convection is a mean over a distribution of times for which members of an ensemble exhibit appreciable convective transport.

Rayleigh–Bénard convection in liquid metal layers under the influence of a horizontal magnetic field

2002 ◽  
Vol 453 ◽  
pp. 345-369 ◽  
Author(s):  
ULRICH BURR ◽  
ULRICH MÜLLER
Keyword(s):  
Magnetic Field ◽  
Heat Transport ◽  
Convective Heat ◽  
Two Dimensional ◽  
Onset Of Convection ◽  
Horizontal Magnetic Field ◽  
Two Dimensional Flow ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard ◽  
The Magnetic Field

This article presents an analytical and experimental study of magnetohydrodynamic Rayleigh–Bénard convection in a large aspect ratio, 20[ratio ]10[ratio ]1, rectangular box. The test fluid is a eutectic sodium potassium Na22K78 alloy with a small Prandtl number of Pr≈0:02. The experimental setup covers Rayleigh numbers in the range 103< Ra<8×104 and Chandrasekhar numbers 0[les ]Q[les ]1.44×106 or Hartmann numbers 0[les ]M[les ]1200, respectively.When a horizontal magnetic field is imposed on a heated liquid metal layer, the electromagnetic forces give rise to a transition of the three-dimensional convective roll pattern into a quasi-two-dimensional flow pattern in such a way that convective rolls become more and more aligned with the magnetic field. A linear stability analysis based on two-dimensional model equations shows that the critical Rayleigh number for the onset of convection of quasi-two-dimensional flow is shifted to significantly higher values due to Hartmann braking at walls perpendicular to the magnetic field. This finding is experimentally confirmed by measured Nusselt numbers. Moreover, the experiments show that the convective heat transport at supercritical conditions is clearly diminished. Adjacent to the onset of convection there is a significant region of stationary convection with significant convective heat transfer before the flow proceeds to time-dependent convection. However, in spite of the Joule dissipation effect there is a certain range of magnetic field intensities where an enhanced heat transfer is observed. Estimates of the local isotropy properties of the flow by a four-element temperature probe demonstrate that the increase in convective heat transport is accompanied by the formation of strong non-isotropic time-dependent flow in the form of large-scale convective rolls aligned with the magnetic field which exhibit a simpler temporal structure compared to ordinary hydrodynamic flow and which are very effective for the convective heat transport.

Effects of Prandtl number in quasi-two-dimensional Rayleigh–Bénard convection

2021 ◽  
Vol 915 ◽  
Author(s):  
Xiao-Ming Li ◽  
Ji-Dong He ◽  
Ye Tian ◽  
Peng Hao ◽  
Shi-Di Huang
Keyword(s):  
Prandtl Number ◽  
Two Dimensional ◽  
Bénard Convection ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

Abstract

Characteristics of high Rayleigh number two-dimensional convection in an open-top porous layer heated from below

1999 ◽  
Vol 394 ◽  
pp. 241-260 ◽  
Author(s):  
ABDELLAH S. M. CHERKAOUI ◽  
WILLIAM S. D. WILCOCK
Keyword(s):  
Rayleigh Number ◽  
Control Volume ◽  
Flow Patterns ◽  
Convective System ◽  
Two Dimensional ◽  
Simpler Algorithm ◽  
Convective Pattern ◽  
Volume Method ◽  
Rayleigh Numbers ◽  
Type Of Transition

Using a control-volume method and the simpler algorithm, we computed steady-state and time-dependent solutions for two-dimensional convection in an open-top porous box, up to a Rayleigh number of 1100. The evolution of the convective system from onset to high Rayleigh numbers is characterized by two types of transitions in the flow patterns. The first type is a decrease in the horizontal aspect ratio of the cells. We observe two such bifurcations. The first occurs at Ra = 107.8 when the convective pattern switches from a steady one-cell roll to a steady two-cell roll. The second occurs at Ra ≈ 510 when an unsteady two-cell roll evolves to a steady four-cell roll. The second type of transition is from a steady to an unsteady pattern and we also observe two of these bifurcations. The first occurs at Ra ≈ 425 in the two-cell convective pattern; the second at Ra ≈ 970 in the four-cell pattern. Both types of bifurcations are associated with an increase in the average vertical convective heat transport. In the bi-cellular solutions, the appearance of non-periodic unsteady convection corresponds to the onset of the expected theoretical scaling Nu ∝ Ra and also to the onset of plume formation. Although our highest quadri-cellular solutions show signs of non-periodic convection, they do not reach the onset of plume formation. An important hysterisis loop exists for Rayleigh numbers in the range 425–505. Unsteady convection appears only in the direction of increasing Rayleigh numbers. In the decreasing direction, steady quadri-cellular flow patterns prevail.

Instabilities of convection rolls in a fluid of moderate Prandtl number

1979 ◽  
Vol 91 (2) ◽  
pp. 319-335 ◽  
Author(s):  
F. H. Busse ◽  
R. M. Clever
Keyword(s):  
Prandtl Number ◽  
Fluid Layer ◽  
Experimental Methods ◽  
Two Dimensional ◽  
Theoretical Predictions ◽  
Convection Rolls ◽  
Rayleigh Numbers ◽  
Horizontal Fluid Layer

The instabilities of two-dimensional convection rolls in a horizontal fluid layer heated from below are investigated in the case when the Prandtl number is seven or lower. Two new mechanisms of instability are described theoretically as well as experimentally. The knot instability causes the transition to spoke-pattern convection at higher Rayleigh numbers while the skewed varicose instability accomplishes a change to larger horizontal wavelengths of the convection rolls. Both instabilities disappear in the limits of small and large Prandtl number. Although the experimental methods fail in realizing closely the infinitely conducting boundaries assumed in the theory, the observations agree in all qualitative aspects with the theoretical predictions.

scholarly journals Two-dimensional magnetohydrodynamic turbulence in the small magnetic Prandtl number limit

2012 ◽  
Vol 703 ◽  
pp. 85-98 ◽  
Author(s):  
David G. Dritschel ◽  
Steven M. Tobias
Keyword(s):  
Magnetic Field ◽  
Prandtl Number ◽  
Spectral Method ◽  
Two Dimensional ◽  
Mhd Turbulence ◽  
Magnetic Prandtl Number ◽  
Wide Range ◽  
Lagrangian Advection ◽  
Pseudo Spectral Method ◽  
Pseudo Spectral

AbstractIn this paper we introduce a new method for computations of two-dimensional magnetohydrodynamic (MHD) turbulence at low magnetic Prandtl number $\mathit{Pm}= \nu / \eta $. When $\mathit{Pm}\ll 1$, the magnetic field dissipates at a scale much larger than the velocity field. The method we utilize is a novel hybrid contour–spectral method, the ‘combined Lagrangian advection method’, formally to integrate the equations with zero viscous dissipation. The method is compared with a standard pseudo-spectral method for decreasing $\mathit{Pm}$ for the problem of decaying two-dimensional MHD turbulence. The method is shown to agree well for a wide range of imposed magnetic field strengths. Examples of problems for which such a method may prove invaluable are also given.

Asymptotic theory of convection in a rotating, cylindrical annulus

1986 ◽  
Vol 173 ◽  
pp. 545-556 ◽  
Author(s):  
F. H. Busse
Keyword(s):  
Equatorial Plane ◽  
Asymptotic Theory ◽  
Rossby Waves ◽  
Numerical Results ◽  
Two Dimensional ◽  
Onset Of Convection ◽  
Rotation Rates ◽  
Cylindrical Annulus ◽  
Simple Equations ◽  
Rayleigh Numbers

The problem of convection in a rotating cylindrical annulus heated from the outside and cooled from the inside is considered in the limit of high rotation rates. The constraint of rotation enforces the two-dimensional character of the motion when the angle of inclination of the axisymmetric end surfaces with respect to the equatorial plane is small. Even when the angle of inclination is large only the dependences on the radial and the azimuthal coordinates need to be considered. The dependence on time at the onset of convection is similar to that of Rossby waves. But at higher Rayleigh numbers a transition to vacillating solutions occurs. In the limit of high rotation rates simple equations can be derived which permit the reproduction and extension of previous numerical results.

Flow Instabilities and Heat Transfer in Buoyancy Driven Flows of Inelastic Non-Newtonian Fluids in Inclined Rectangular Cavities

2010 ◽  
Author(s):  
Dennis Siginer ◽  
Lyes Khezzar
Keyword(s):  
Heat Transfer ◽  
Prandtl Number ◽  
Heat Transfer Rate ◽  
Power Law ◽  
Transfer Rate ◽  
Newtonian Fluids ◽  
Two Dimensional ◽  
Aspect Ratios ◽  
Flow Configurations ◽  
Rayleigh Numbers

Steady two-dimensional natural convection in rectangular two dimensional cavities filled with non-Newtonian power law-Boussinesq fluids is numerically investigated. The conservation equations of mass, momentum and energy are solved using the finite volume method for varying inclination angles between 0° and 90° and two cavity height based Rayleigh numbers, Ra = 104 and 105, a Prandtl number of Pr = 102 and two cavity aspect ratios of 1, 4. For the vertical inclination of 90°, computations were performed for two Rayleigh numbers Ra = 104 and 105 and three Prandtl numbers of Pr = 102, 103 and 104. In all of the numerical experiments, the channel is heated from below and cooled from the top with insulated side-walls and the inclination angle is varied. A comprehensive comparison between the Newtonian and the non-Newtonian cases is presented based on the dependence of the average Nusselt number Nu on the angle of inclination together with the Rayleigh number, Prandtl number, power law index n and aspect ratio dependent flow configurations which undergo several exchange of stability as the angle of inclination O̸ is gradually increased from the horizontal resulting in a rather sudden drop in the heat transfer rate triggered by the last loss of stability and transition to a single cell configuration. Despite significant differences in the heat transfer rate and flow configurations both Newtonian and non-Newtonian fluids of the power law type exhibit qualitatively similar behavior.

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