Non-Boussinesq simulations of Rayleigh–Bénard convection in a perfect gas (original) (raw)
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Physics of Fluids, 2013
We present the results of direct numerical simulations of flow patterns in a low-Prandtl-number (P r = 0.1) fluid above the onset of oscillatory convection in a Rayleigh-Bénard system rotating uniformly about a vertical axis. Simulations were carried out in a periodic box with thermally conducting and stress-free top and bottom surfaces. We considered a rectangular box (L x ×L y ×1) and a wide range of Taylor numbers (750 ≤ T a ≤ 5000) for the purpose. The horizontal aspect ratio η = L y /L x of the box was varied from 0.5 to 10. The primary instability appeared in the form of two-dimensional standing waves for shorter boxes (0.5 ≤ η < 1 and 1 < η < 2). The flow patterns observed in boxes with η = 1 and η = 2 were different from those with η < 1 and 1 < η < 2. We observed a competition between two sets of mutually perpendicular rolls at the primary instability in a square cell (η = 1) for T a < 2700, but observed a set of parallel rolls in the form of standing waves for T a ≥ 2700. The three-dimensional convection was quasiperiodic or chaotic for 750 ≤ T a < 2700, and then bifurcated into a two-dimensional periodic flow for T a ≥ 2700. The convective structures consisted of the appearance and disappearance of straight rolls, rhombic patterns, and wavy rolls inclined at an angle φ = π 2 − arctan (η −1) with the straight rolls.
Oscillating Instability in One-Dimensional Rayleigh-Benard Convection
Le Journal de Physique Colloques, 1989
-Une oscillation collective périodique de la position des rouleaux est observée au cours d'expériences de convection de Rayleigh-Bénard en géométrie quasi-unidimensionnelle-c.a.d. quand l'une des dimensions horizontales est inférieure à l'épaisseur de la couche de fluide. Ces oscillations sont liées à la présence de très petites longueurs d'onde et conduisent directement à des comportements d'intermittences turbulentes spatio temporelles lorsque la géométrie est annulaire. Abstract-Collective periodic oscillations of the rolls' position are observed in experiments of Rayleigh-Benard convection, when one of the horizontal extensions is smaller than the depth of the fluid layer (quasi one dimensional geometry). The oscillations are related to the presence of very short wavelengths and can lead directly to turbulent spatio temporal intermittencies, when the convection is achieved in an annular geometry.
Oscillatory instabilities of convection rolls at intermediate Prandtl numbers
Journal of Fluid Mechanics, 1986
The analysis of the instabilities of convection rolls in a fluid layer heated from below with no-slip boundaries exhibits a close competition between various oscillatory modes in the range 2 [lsim ] P [lsim ] 12 of the Prandtl number P. In addition to the even-oscillatory instability known from earlier work two new instabilities have been found, each of which is responsible for a small section of the stability boundary of steady rolls. The most interesting property of the new instabilities is their close relationship to the hot-blob oscillations known from experimental studies of convection. In the lower half of the Prandtl-number range considered the B02-mode dominates, which is characterized by two blobs each of slightly hotter and colder fluid circulating around in the convection roll in a spatially and time-periodic fashion. At higher Prandtl numbers the BE 1-mode dominates, which possesses one hot blob (and one cold blob) circulating with the convection velocity. Just outside t...
Instability in temperature modulated rotating Rayleigh–B ´ enard convection
The problem of instability in an infinite horizontal thin layer of a Boussinesq fluid, uniformly rotated and heated from below with time periodic oscillations of the wall temperatures is investigated theoretically and numerically. In doing so, modest rotation rate is considered such that the Froude number does not exceed 0.05 which allows neglect of centrifugal effects. The Floquet analysis is used to obtain the critical Rayleigh number as a function of the parameters controlling the system. The instability is found to manifest itself in the form of a flow which oscillates either harmonically or subharmonically depending upon the control parameters. Instability regions in an appropriate parametric space of the dimensionless wave number of disturbance imposed over the flow and the dimensionless amplitude of modulation, at the onset of time periodic fluid flows are obtained numerically. The modulation amplitude, the modulation frequency and the rotation rate, are observed to affect the stability of the flow considerably. (Some figures may appear in colour only in the online journal)
Physical Review E, 2004
Motivated by the Küppers-Lortz instability of roll patterns in the presence of rotation, we have investigated the effects of rotation on a hexagonal pattern in Rayleigh-Bénard convection. While several theoretical models have been developed, experimental data cannot be found in the literature. In order to check the validity of the predictions and to study the effects of rotation on the behavior of the system, we present experimental results for a non-Boussinesq Rayleigh-Bénard convection with rotation about the vertical axis. Rotation introduces an additional control parameter, namely the dimensionless rotation rate ⍀ =2fd 2 / , where f is the rotation rate (in Hz), d is the thickness of the cell, and is the kinematic viscosity. We observe that the cell rotation induces a slow rotation of the pattern in the opposite direction ͑Ϸ⍀ϫ10 −4 ͒ in the rotating frame. Moreover, it tends to destroy the convective pattern. No oscillation of the hexagonal pattern over the range of its existence ͑⍀ഛ6͒ has been observed.
Journal of Fluid Mechanics, 2005
Nonlinear solutions in the form of squares and rolls are investigated for Rayleigh-Bénard convection in an infinite-Prandtl-number fluid enclosed between two symmetric slabs. It is found that the heat transfer depends strongly on the thickness and thermal conductivity of the slabs, but hardly on the planform of convection. Examples of stability regions of rolls are calculated, showing that for certain slab selections, rolls remain stable at even larger Rayleigh numbers than with fixed temperatures at the boundaries. The region of stable squares is restricted by a zigzag and a longwavelength cross-roll instability in addition to a new three-dimensional instability. As the slab conductivity is increased, the stability region of the squares shrinks onto a point located well above the critical point for the onset of convection. For a small range of slab conductivities, stability regions for squares and rolls both exist for the same set-up. In the present calculations, the regions never overlap. An example, where both patterns are stable at the same Rayleigh number, provides an explanation for the co-existence of rolls and squares where transparent slabs with a low thermal conductivity were applied.
Stability of convection rolls in a layer with stress-free boundaries
Journal of Fluid Mechanics, 1985
Steady finite-amplitude solutions for two-dimensional convection in a layer heated from below with stress-free boundaries are obtained numerically by a Galerkin method. The stability of the steady convection rolls with respect to arbitrary three-dimensional infinitesimal disturbances is investigated. Stability is found only in a small fraction of the Rayleigh-number-wavenumber space where steady solutions exist. The cross-roll instability and the oscillatory and monotonic skewed varicose instabilities are most important in limiting the stability of steady convection rolls. The Prandtlnumbers P = 0.71, 7, 104 areemphasized, but the stability boundaries are sufficiently smoothly dependent on the parameters of the problem to permit qualitative extrapolations to other Prandtl numbers.
Dynamics of defects in Rayleigh-Bénard convection
Physical Review A, 1981
The behavior of an extra roll extending into an otherwise regular convection pattern is studied as a function of Rayleigh number, Prandtl number, P, and wavelength, by means of a fully resolved numerical simulation of the Boussinesq equations with free-slip boundary conditions. For .reduced Rayleigh numbers of order one or less and P &40, numerical simulations of the lowest-order amplitude equations reproduce the Boussinesq results semiquantitatively. In particular, we find that when this class of defects is stable, they move with constant velocity v, parallel to the roll axis and give rise to a slow modulation of the roll pattern of the form f(x,yvt). Both f and v have been calculated analytically within a linearized theory. The envelope function f depends in an essential way on v such that the limit v~0 cannot be sensibly taken.
Physics of Fluids, 2011
Experimental and numerical studies of steady longitudinal convection rolls that develop in a Poiseuille air flow in a rectangular channel heated from below and cooled from the top are conducted in the range 3500 Ra 6000 and 20 Re 200. The effect of the lateral vertical walls on the onset and development of the convection cells is investigated by changing the transverse aspect ratio of the channel from 4.7 to 18.4. The influence of the entrance temperature and adiabatic or conductive thermal boundary conditions at the side and top walls of the channel is also investigated. The scenario of the roll formation is described in details. It results in a symmetric pattern in the form of steady longitudinal rolls with an even number of rolls that depends not only on the aspect ratio but possibly on the inlet temperature of the flow. It is shown that the fully developed pattern is determined by the two rolls nearby each vertical side wall that are triggered just at the entrance of the channel due to the presence of velocity boundary layers adjacent to the walls. It is also shown that the heat conduction in the top horizontal wall of the experimental channel must be taken into account in the numerical simulations so that the experimental wavenumber can be properly depicted.
Multiple modes of instability in a box heated from the side in low-Prandtl-number fluids
Physics of Fluids, 2007
The existence of multiple modes of instability in Rayleigh-Bénard or Marangoni-Bénard situations has been known for many years. This existence is shown for the first time for low-Prandtl-number flows in three-dimensional cavities heated from the side. For such a situation, the study of the flow transitions has long remained a challenge, as these transitions occur in already very intense flows. The study is possible here thanks to performing numerical methods, and the ten first instability modes are determined for a wide range of aspect ratios and Prandtl number values. The most striking feature of our results is the very frequent change of leading mode when aspect ratios or Prandtl number are changed, which indicates different flow structures triggered at the transitions, either steady or oscillatory and breaking some of the symmetries of the problem.