Asymptotically exact codimension-four dynamics and bifurcations in two-dimensional thermosolutal convection at high thermal Rayleigh number: Chaos from a quasi-periodic homoclinic explosion and quasi-periodic intermittency

2018 ◽  
Vol 382-383 ◽  
pp. 1-21 ◽  
Author(s):  
Justin S. Eilertsen ◽  
Jerry F. Magnan
Keyword(s):  
Rayleigh Number ◽  
Thermosolutal Convection ◽  
Two Dimensional ◽  
Homoclinic Explosion ◽  
Thermal Rayleigh Number

Oscillations in double-diffusive convection

1981 ◽  
Vol 109 ◽  
pp. 25-43 ◽  
Author(s):  
L. N. Da Costa ◽  
E. Knobloch ◽  
N. O. Weiss
Keyword(s):  
Rayleigh Number ◽  
Period Doubling ◽  
Thermosolutal Convection ◽  
Two Dimensional ◽  
Double Diffusive Convection ◽  
Dimensional Problem ◽  
Diffusive Convection ◽  
Steady Solutions ◽  
Steady Convection ◽  
Double Diffusive

We have studied the transition between oscillatory and steady convection in a simplified model of two-dimensional thermosolutal convection. This model is exact to second order in the amplitude of the motion and is qualitatively accurate for larger amplitudes. If the ratio of the solutal diffusivity to the thermal diffusivity is sufficiently small and the solutal Rayleigh number, RS, sufficiently large, convection sets in as overstable oscillations, and these oscillations grow in amplitude as the thermal Rayleigh number, RT, is increased. In addition to this oscillatory branch, there is a branch of steady solutions that bifurcates from the static equilibrium towards lower values of RT; this subcritical branch is initially unstable but acquires stability as it turns round towards increasing values of RT. For moderate values of RS the oscillatory branch ends on the unstable (subcritical) portion of the steady branch, where the period of the oscillations becomes infinite. For larger values of RS a birfurcation from symmetrical to asymmetrical oscillations is followed by a succession of bifurcations, at each of which the period doubles, until the motion becomes aperiodic at some finite value of RT. The chaotic solutions persist as RT is further increased but eventually they lose stability and there is a transition to the stable steady branch. These results are consistent with the behaviour of solutions of the full two-dimensional problem and suggest that period-doubling, followed by the appearance of a strange attractor, is a characteristic feature of double-diffusive convection.

scholarly journals Two-Dimensional Convection Induced by Gravity and Centrifugal Forces in a Rotating Porous Layer Far Away from the Axis of Rotation

1998 ◽  
Vol 4 (2) ◽  
pp. 73-90 ◽  
Author(s):  
Peter Vadasz ◽  
Saneshan Govender
Keyword(s):  
Rayleigh Number ◽  
Porous Layer ◽  
Temperature Fields ◽  
Wave Number ◽  
Marginal Stability ◽  
Two Dimensional ◽  
Wide Range ◽  
Gravity Driven ◽  
Gravity Driven Flow ◽  
The Stability

The stability and onset of two-dimensional convection in a rotating fluid saturated porous layer subject to gravity and centrifugal body forces is investigated analytically. The problem corresponding to a layer placed far away from the centre of rotation was identified as a distinct case and therefore justifying special attention. The stability of a basic gravity driven convection is analysed. The marginal stability criterion is established in terms of a critical centrifugal Rayleigh number and a critical wave number for different values of the gravity related Rayleigh number. For any given value of the gravity related Rayleigh number there is a transitional value of the wave number, beyond which the basic gravity driven flow is stable. The results provide the stability map for a wide range of values of the gravity related Rayleigh number, as well as the corresponding flow and temperature fields.

The Influence of Real Gas Radiation On the Stability and Development of Benard Convection in a Two-Dimensional Layer

2021 ◽  
Author(s):  
Brent W. Webb ◽  
Vladimir Solovjov
Keyword(s):  
Rayleigh Number ◽  
Critical Rayleigh Number ◽  
Radiative Heating ◽  
Two Dimensional ◽  
Gas Temperature ◽  
Layer Depth ◽  
Real Gas ◽  
Gas Radiation ◽  
Induced Flow ◽  
Buoyancy Induced Flow

Abstract The influence of real gas radiation on the thermal and hydrodynamic stability is investigated in a two-dimensional layer of radiatively participating H2O and/or CO2 heated from below. The non-gray radiation effects of the two species are treated rigorously using a global spectral approach, the Spectral Line Weighted-sum-of-gray-gases model. The phenomena are explored by solving the full coupled laminar equations of motion, energy, and radiative transfer from the low-Rayleigh number, pure conduction-radiation regime through the onset of buoyancy-induced flow to the developed Bénard convection regime. The evolution of the thermal, velocity, and radiative heating fields is studied, and the critical Rayleigh number is characterized as a function of species mole fraction, average layer gas temperature, layer depth, wall emissivity, and the total gas pressure. It is found that participating radiation in the medium has the effect of stabilizing the layer, delaying transition to buoyancy-induced flow. The development of buoyancy-induced flow and temperature, along with the radiative heating are presented. It is found that the critical Rayleigh number in the radiatively participating gas layer can be more than an order of magnitude higher than the classical convection-only scenario. The onset of instability is found to depend on the species mole fractions, average gas temperature in the layer, wall emissivity, layer depth, and total pressure. Generally, all other variables being the same, H2O has a greater stabilizing influence on the layer than CO2.

scholarly journals Linear biglobal analysis of Rayleigh–Bénard instabilities in binary fluids with and without throughflow

2012 ◽  
Vol 713 ◽  
pp. 216-242 ◽  
Author(s):  
Jun Hu ◽  
Daniel Henry ◽  
Xie-Yuan Yin ◽  
Hamda BenHadid
Keyword(s):  
Reynolds Number ◽  
Rayleigh Number ◽  
Aspect Ratio ◽  
Critical Rayleigh Number ◽  
Two Dimensional ◽  
Binary Fluids ◽  
Transverse Modes ◽  
Aspect Ratios ◽  
Rayleigh Benard ◽  
Mode A

AbstractThree-dimensional Rayleigh–Bénard instabilities in binary fluids with Soret effect are studied by linear biglobal stability analysis. The fluid is confined transversally in a duct and a longitudinal throughflow may exist or not. A negative separation factor $\psi = \ensuremath{-} 0. 01$, giving rise to oscillatory transitions, has been considered. The numerical dispersion relation associated with this stability problem is obtained with a two-dimensional Chebyshev collocation method. Symmetry considerations are used in the analysis of the results, which allow the classification of the perturbation modes as ${S}_{l} $ modes (those which keep the left–right symmetry) or ${R}_{x} $ modes (those which keep the symmetry of rotation of $\lrm{\pi} $ about the longitudinal mid-axis). Without throughflow, four dominant pairs of travelling transverse modes with finite wavenumbers $k$ have been found. Each pair corresponds to two symmetry degenerate left and right travelling modes which have the same critical Rayleigh number ${\mathit{Ra}}_{c} $. With the increase of the duct aspect ratio $A$, the critical Rayleigh numbers for these four pairs of modes decrease and closely approach the critical value ${\mathit{Ra}}_{c} = 1743. 894$ obtained in a two-dimensional situation, one of the mode (a ${S}_{l} $ mode called mode A) always remaining the dominant mode. Oscillatory longitudinal instabilities ($k\approx 0$) corresponding to either ${S}_{l} $ or ${R}_{x} $ modes have also been found. Their critical curves, globally decreasing, present oscillatory variations when the duct aspect ratio $A$ is increased, associated with an increasing number of longitudinal rolls. When a throughflow is applied, the symmetry degeneracy of the pairs of travelling transverse modes is broken, giving distinct upstream and downstream modes. For small and moderate aspect ratios $A$, the overall critical Rayleigh number in the small Reynolds number range studied is only determined by the upstream transverse mode A. In contrast, for larger aspect ratios as $A= 7$, different modes are successively dominant as the Reynolds number is increased, involving both upstream and downstream transverse modes A and even the longitudinal mode.

Convection fluid dynamics in a model of a fault zone in the earth's crust

1978 ◽  
Vol 88 (4) ◽  
pp. 769-792 ◽  
Author(s):  
D. R. Kassoy ◽  
A. Zebib
Keyword(s):  
Fluid Dynamics ◽  
Rayleigh Number ◽  
Porous Material ◽  
Fault Zone ◽  
Slug Flow ◽  
Tectonic Activity ◽  
Two Dimensional ◽  
Core Flow ◽  
Number Flow ◽  
Large Rayleigh Number

Faulted regions associated with geothermal areas are assumed to be composed of rock which has been heavily fractured within the fault zone by continuous tectonic activity. The fractured zone is modelled as a vertical, slender, two-dimensional channel of saturated porous material with impermeable walls on which the temperature increases linearly with depth. The development of an isothermal slug flow entering the fault at a large depth is examined. An entry solution and the subsequent approach to the fully developed configuration are obtained for large Rayleigh number flow. The former is characterized by growing thermal boundary layers adjacent to the walls and a slightly accelerated isothermal core flow. Further downstream the development is described by a parabolic system. It is shown that a class of fully developed solutions is not spatially stable.

Natural Convection in a Micro Size Horizontal Cylindrical Annulus With Periodic Volumetric Heat Flux

2009 ◽  
Author(s):  
Kamyar Mansour
Keyword(s):  
Heat Flux ◽  
Natural Convection ◽  
Rayleigh Number ◽  
Viscous Fluid ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Similarity Parameter ◽  
Symbolic Calculation ◽  
Cylindrical Annulus ◽  
Gap Ratio

We consider the two-dimensional problem of steady natural convection in a narrow (Micro size) Horizontal Cylindrical annulus filled with viscous fluid and periodic volumetric heat flux. The solution is expanded in powers of a single combined similarity parameter, which is the product of the Gap ratio to the power of four, and Rayleigh number and the series extended by means of symbolic calculation up to 16 terms. Analysis of these expansions allows the exact computation for arbitrarily accuracy up to 50000 figures. Although the range of the radius of convergence is almost zero but Pade approximation lead our result to be good even for much higher value of the similarity parameter.

Symbolic Calculation for Free Convection in a Circular Cavity With Periodic Heat Generation

2009 ◽  
Author(s):  
Kamyar Mansour
Keyword(s):  
Natural Convection ◽  
Rayleigh Number ◽  
Free Convection ◽  
Heat Generation ◽  
Circular Cavity ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Similarity Parameter ◽  
Symbolic Calculation ◽  
Periodic Heat

We consider the two-dimensional problem of steady natural convection in a circular cavity with periodic heat generation filled with viscous fluid subject to cosine temperature variation on the boundary. The solution is expanded for low Rayleigh number and extended to 16 terms by computer. Analysis of these expansions allows the exact computation for arbitrarily accuracy up to 50000 figures. Although the range of the radius of convergence is small but pade approximation leads our result to be good even for higher value of the similarity parameter.

Free Convection in a Two-Dimensional Porous Loop

1986 ◽  
Vol 108 (2) ◽  
pp. 277-283 ◽  
Author(s):  
L. Robillard ◽  
T. H. Nguyen ◽  
P. Vasseur
Keyword(s):  
Steady State ◽  
Rayleigh Number ◽  
Porous Layer ◽  
Outer Boundary ◽  
Convective Motion ◽  
Steady State Solution ◽  
Temperature Fields ◽  
Maximum Temperature ◽  
Two Dimensional ◽  
Convective Cells

A study is made of the natural convection in an annular porous layer having an isothermal inner boundary and its outer boundary subjected to a thermal stratification arbitrarily oriented with respect to gravity. For such conditions, no symmetry can be expected for the flow and temperature fields with respect to the vertical diameter and the whole circular region must be considered. Two-dimensional steady-state solutions are sought by perturbation and numerical approaches. Results obtained indicate that the circulating flow around the annulus attains its maximum strength when the stratification is horizontal (heating from the side). This circulating flow is responsible for an important heat exchange between the porous layer and its external surroundings. The flow field is also characterized by the presence of two convective cells near the inner boundary, giving rise to flow reversal on this surface. When the maximum temperature on the outer boundary is at the bottom of the cavity, the convective motion becomes potentially unstable; for a Rayleigh number below 80, there exists a steady-state solution symmetrical with respect to both vertical and horizontal axes; for a Rayleigh number above 80, an unsteady periodic situation develops with the circulating flow alternating its direction around the annulus.

Convection in an imposed magnetic field. Part 1. The development of nonlinear convection

1981 ◽  
Vol 108 ◽  
pp. 247-272 ◽  
Author(s):  
N. O. Weiss
Keyword(s):  
Magnetic Field ◽  
Thermal Diffusivity ◽  
Rayleigh Number ◽  
Numerical Experiments ◽  
Dynamical Regime ◽  
Two Dimensional ◽  
Nonlinear Convection ◽  
Boussinesq Fluid ◽  
Chandrasekhar Number ◽  
Steady Convection

Nonlinear two-dimensional magnetoconvection in a Boussinesq fluid has been studied in a series of numerical experiments with values of the Chandrasekhar number Q ≤ 4000 and the ratio ζ of the magnetic to the thermal diffusivity in the range 1 ≥ ζ ≥ 0·025. If the imposed field is strong enough, convection sets in as overstable oscillations which give way to steady convection as the Rayleigh number R is increased. In the dynamical regime that follows, magnetic flux is concentrated into sheets at the sides of the cells, from which the motion is excluded.

Transition and turbulence in horizontal convection: linear stability analysis

2017 ◽  
Vol 821 ◽  
pp. 31-58 ◽  
Author(s):  
Pierre-Yves Passaggia ◽  
Alberto Scotti ◽  
Brian White
Keyword(s):  
Stability Analysis ◽  
Rayleigh Number ◽  
Boundary Conditions ◽  
Stokes Equations ◽  
Critical Rayleigh Number ◽  
Base Flow ◽  
Two Dimensional ◽  
Slip Boundary Conditions ◽  
Slip Boundary ◽  
Horizontal Convection

The linear instability mechanisms of horizontal convection in a rectangular cavity forced by a horizontal buoyancy gradient along its surface are investigated using local and global stability analyses for a Prandtl number equal to unity. The results show that the stability of the base flow, a steady circulation characterized by a narrow descending plume and a broad upwelling region, depends on the Rayleigh number, $Ra$. For free-slip boundary conditions at a critical value of $Ra\approx 2\times 10^{7}$, the steady base flow becomes unstable to three-dimensional perturbations, characterized by counter-rotating vortices originating within the plume region. A Wentzel–Kramers–Brillouin (WKB) method applied along closed streamlines demonstrates that this instability is of a Rayleigh–Taylor type and can be used to accurately reconstruct the global instability mode. In the case of no-slip boundary conditions, the base flow also becomes unstable to a self-sustained two-dimensional instability whose critical Rayleigh number is $Ra\approx 1.7\times 10^{8}$. Beyond this critical $Ra$, two-dimensional equilibrium stationary states of the Navier–Stokes equations are computed using the selective frequency damping method. The two-dimensional onset of instability is shown to be characterized by a family of modes also originating within the plume. A local spatio-temporal stability analysis shows that the flow becomes absolutely unstable at the origin of the plume. Taken together, these results illustrate the mechanisms behind the onset of turbulence that has been observed in horizontal convection.

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