Buoyancy-Driven Flow Transitions in Deep Cavities Heated From Below

2002 ◽  
Vol 124 (4) ◽  
pp. 650-659 ◽  
Author(s):  
Chunmei Xia ◽  
Jayathi Y. Murthy
Keyword(s):  
Rayleigh Number ◽  
Oscillatory Flow ◽  
Critical Rayleigh Number ◽  
Three Dimensional ◽  
Chaotic Flow ◽  
Aspect Ratios ◽  
Transition To Chaos ◽  
Route To Chaos ◽  
Flow Transitions ◽  
Deep Cavities

A numerical investigation has been conducted of flow transitions in deep three-dimensional cavities heated from below. The first critical Rayleigh number, RaI, below which the flow is at rest, and the second critical Rayleigh number, RaII, for transition from steady state to oscillatory flow, have been found for cavities of aspect ratios Ar in the range 1–5. Transition to chaos has also been examined for these cases. The results show that RaI=3583,2.545×104 and 5.5×105 and RaII=4.07×105,1.65×106 and 1.30×107 for aspect ratios of 1, 2, and 5 respectively. The route to chaos is PPeriodic→QP2(Quasi-periodic with two incommensurate frequencies)→QP3(Quasi-periodic with three incommensurate frequencies)→NChaotic for Ar=1 with the Rayleigh number varying from 4.07×105 to 4.89×105. The route is PPeriodic→P2(Periodic doubling)→I(Intermittent)→P(Periodic)→N(Chaotic) for Ar=2 over a Ra range of 1.65×106 to 1.83×106. The interval between periodic and chaotic flow is very short for Ar=5.

Three-dimensional instability of axisymmetric buoyant convection in cylinders heated from below

1996 ◽  
Vol 326 ◽  
pp. 399-415 ◽  
Author(s):  
M. Wanschura ◽  
H. C. Kuhlmann ◽  
H. J. Rath
Keyword(s):  
Rayleigh Number ◽  
Linear Stability ◽  
Critical Rayleigh Number ◽  
Three Dimensional ◽  
Base Flow ◽  
Onset Of Convection ◽  
Buoyant Convection ◽  
Thermal Boundary Conditions ◽  
Aspect Ratios ◽  
Prandtl Numbers

The stability of steady axisymmetric convection in cylinders heated from below and insulated laterally is investigated numerically using a mixed finite-difference/Chebyshev collocation method to solve the base flow and the linear stability equations. Linear stability boundaries are given for radius to height ratios γ from 0.9 to 1.56 and for Prandtl numbers Pr = 0.02 and Pr = 1. Depending on γ and Pr, the azimuthal wavenumber of the critical mode may be m = 1, 2, 3, or 4. The dependence of the critical Rayleigh number on the aspect ratio and the instability mechanisms are explained by analysing the energy transfer to the critical modes for selected cases. In addition to these results the onset of buoyant convection in liquid bridges with stress-free conditions on the cylindrical surface is considered. For insulating thermal boundary conditions, the onset of convection is never axisymmetric and the critical azimuthal wavenumber increases monotonically with γ. The critical Rayleigh number is less then 1708 for most aspect ratios.

The development of asymmetry and period doubling for oscillatory flow in baffled channels

1996 ◽  
Vol 328 ◽  
pp. 19-48 ◽  
Author(s):  
E. P. L. Roberts ◽  
M. R. Mackley
Keyword(s):  
Reynolds Number ◽  
Oscillatory Flow ◽  
Three Dimensional ◽  
Period Doubling ◽  
Progressive Increase ◽  
Newtonian Flow ◽  
Chaotic Flow ◽  
Spatially Periodic ◽  
Dimensional Simulation ◽  
Time Periodic

We report experimental and numerical observations on the way initially symmetric and time-periodic fluid oscillations in baffled channels develop in complexity. Experiments are carried out in a spatially periodic baffled channel with a sinusoidal oscillatory flow. At modest Reynolds number the observed vortex structure is symmetric and time periodic. At higher values the flow progressively becomes three-dimensional, asymmetric and aperiodic. A two-dimensional simulation of incompressible Newtonian flow is able to follow the flow pattern at modest oscillatory Reynolds number. At higher values we report the development of both asymmetry and a period-doubling cascade leading to a chaotic flow regime. A bifurcation diagram is constructed that can describe the progressive increase in complexity of the flow.

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 in a box: linear theory

1967 ◽  
Vol 30 (3) ◽  
pp. 465-478 ◽  
Author(s):  
Stephen H. Davis
Keyword(s):  
Rayleigh Number ◽  
Linear Stability ◽  
Linear Theory ◽  
Cross Sections ◽  
Critical Rayleigh Number ◽  
Three Dimensional ◽  
Wave Number ◽  
Spatial Variables ◽  
Zero Velocity ◽  
The Cross

The linear stability of a quiescent, three-dimensional rectangular box of fluid heated from below is considered. It is found that finite rolls (cells with two non-zero velocity components dependent on all three spatial variables) with axes parallel to the shorter side are predicted. When the depth is the shortest dimension, the cross-sections of these finite rolls are near-square, but otherwise (in wafer-shaped boxes) narrower cells appear. The value of the critical Rayleigh number and preferred wave-number (number of finite rolls) for a given size box is determined for boxes with horizontal dimensions h, ¼ ≤ h/d ≤ 6, where d is the depth.

Numerical Simulations of an Unsteady Confined Natural Convection Flow

2009 ◽  
Author(s):  
Gillian Leplat ◽  
Emmanuel Laroche ◽  
Philippe Reulet ◽  
Pierre Millan
Keyword(s):  
Natural Convection ◽  
Rayleigh Number ◽  
Large Scale ◽  
Critical Rayleigh Number ◽  
Three Dimensional ◽  
High Amplitude ◽  
Joint Fluid ◽  
Convection Flow ◽  
Two Dimensional ◽  
Natural Convection Flow

A two-dimensional numerical analysis of a laminar natural convection flow within an air-filled enclosure is proposed in this paper from an unstable configuration previously studied experimentally. The flow is driven by a heated square-section cylinder located at the center of a square-section enclosure. Instabilities are observed for an aspect ratio (height of the cylinder over the height of the cavity) of 0.4 and cause the flow to turn into a three-dimensional and unsteady regime characterized by a symmetry breaking and large scale high amplitude flappings around the cylinder. The multi-physic computational software CEDRE, developed at the ONERA, is used to study this unstable behavior and a time-dependent compressible flow solver is used to perform the two-dimensional simulations under the low Mach number approximation, corresponding to the mid-depth cross-section of the enclosure from the experimental configuration. The first results on the investigation of the first unstable modes confirm the onset of the instabilities at the Rayleigh number of the experiment with asymmetrical motions of the fluid around the cylinder. Further analyses highlight the critical Rayleigh number that defines the instability threshold of the first bifurcation which origin and nature could have been identified. Finally, joint fluid-solid simulations are performed to determine more precisely the role of boundary conditions in the onset of instabilities.

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.

Enabling Resist Processing Technologies for Advanced Packaging

2014 ◽  
Vol 2014 (DPC) ◽  
pp. 000830-000862 ◽  
Author(s):  
Antun Peic ◽  
Thorsten Matthias ◽  
Johanna Bartl ◽  
Paul Lindner
Keyword(s):  
Three Dimensional ◽  
Design Solution ◽  
Wafer Level ◽  
Coating Technology ◽  
3D Ic ◽  
Wafer Level Packaging ◽  
Aspect Ratios ◽  
Integration Schemes ◽  
Interconnect Design ◽  
Deep Cavities

The increasing adoption of advanced wafer-level packaging (WLP) technologies and high density interposer concepts clearly reflect the permanent need for form factor reduction, smaller process geometries and higher-count I/O on ICs. Currently, several strategies are being pursued to achieve these goals. The most promising approaches are summarized under the concept of three-dimensional integrated circuits (3D-IC) and three-dimensional wafer level packaging (3D-WLP) technology. A key component for 3D device integration schemes is the requirement of vertical through-silicon-via (TSV) interconnections that enables electrical through-chip communication through stacks of vertically integrated layers on the wafer scale. Ultimately, the use of TSVs also enables higher performance and smaller package sizes. In order to realize TSV connections, a series of process steps is required such as the thinning and bonding of the wafer to a carrier prior to the formation of through-wafer vias, followed by the passivation and metallization of the vias. Despite the potential benefits associated with the integration of TSVs also significant challenges have to be overcome. One of the greatest challenges for present and even more for upcoming TSV design strategies still remains the processing of photoresist and other functional polymers at and within TSV geometries. To this day, it is still very difficult to achieve a conformal polymer coating in deep cavities, along steep side walls and especially within the extreme aspect ratios of TSV. Mainly this is due to the fact that standard surface coating methods such as spin coating were just not developed to meet the requirements posed by these high aspect ratio microstructures. New and innovative approaches are needed to meet these new challenges. Spray coating is one of the most promising technologies to overcome current barriers. However, even most of the available spray deposition equipment is facing its limits with steadily decreasing via diameters and increasing aspect ratios on the other hand. Successively, the multitude of these challenging technological developments in the 3D-IC and wafer-level packaging area has created the demand for innovative manufacturing approaches, new equipment and related tools. Herein we present our new EVG ®NanoSprayTM coating technology with unique capabilities to overcome the present limits of conformal resist coating over extreme topography. We demonstrate one particularly promising application for conformal polymer coatings; as an annular lining at the interface between the conducting metal filling in the TSV and the silicon wafer. The intrinsic properties of the polymer allow a TSV design solution that is more forgiving on coefficient of thermal expansion (CTE) mismatch-induced stress between the silicon substrate and the interfacing metal. Consequently, this new type of polymer buffered TSV interconnect design promises to significantly reduce thermal stress-induced TSV delamination as one of the dominant failure modes for 3-D interconnects. We further demonstrate the application of EVG ®NanoSprayTM as enabling coating technology for llithographic processing of conformal coated TSVs. The patterning of thin photoresist layers at the bottom of vias and along the steep sidewalls of deep cavities allows for more degrees of freedom in electrical contact formation. The presented EVG ®NanoSprayTM coating technology opens new dimensions in advanced wafer level packaging and provokes reconsidering prevailing limitations in interconnect design.

Numerical simulations of multiple flow transitions in axisymmetric annulus convection

1989 ◽  
Vol 206 ◽  
pp. 517-544 ◽  
Author(s):  
P. Le Quéré ◽  
J. Pécheux
Keyword(s):  
Hopf Bifurcation ◽  
Rayleigh Number ◽  
Travelling Waves ◽  
Critical Rayleigh Number ◽  
Period Doubling ◽  
Boussinesq Equations ◽  
Time Stepping ◽  
Nusselt Numbers ◽  
Steady Oscillations ◽  
Flow Transitions

We examine numerically the behaviour of the solutions of the axisymmetric Boussinesq equations in a tall, differentially heated, air-filled annulus. The numerical algorithm integrates the time-dependent equations in primitive variables and combines a pseudospectral Chebyshev spatial expansion with a second-order time-stepping scheme. The instability of the conduction regime is found to be unsteady cross-rolls. By assuming Hopf bifurcation, we can accurately determine the critical Rayleigh number. As the Rayleigh number increases, the solution is monoperiodic at first. Then it undergoes a period-doubling bifurcation. When the Rayleigh number is further increased, the solution reverts to a monocellular steady state through suberitical bifurcations with hysteresis. At even higher Rayleigh number, boundary-layer instability sets in, in the form of travelling waves. This instability has the characteristics of a supercritical Hopf bifurcation. We examine the space-time structure of the two types of unsteady solutions. We have presented the basic periods of the steady oscillations as functions of the Rayleigh number in the vicinity of the Hopf bifurcation points, and have also computed the Nusselt numbers for the various flow regimes.

Rayleigh-Be´nard Convection in a Small Aspect Ratio Enclosure: Part II—Bifurcation to Chaos

1993 ◽  
Vol 115 (2) ◽  
pp. 367-376 ◽  
Author(s):  
D. Mukutmoni ◽  
K. T. Yang
Keyword(s):  
Numerical Study ◽  
Period Doubling ◽  
Small Aspect Ratio ◽  
Mean Flow ◽  
Average Temperature ◽  
Aspect Ratios ◽  
Transition To Chaos ◽  
The Mean ◽  
Route To Chaos ◽  
Boussinesq Fluid

The present numerical study documents bifurcation sequences for Rayleigh-Be´nard convection in a rectangular enclosure with insulated sidewalls. The aspect ratios are 3.5 and 2.1 and the Boussinesq fluid is water (average temperature of 70°C) with a Prandtl number of 2.5. The transition to chaos observed in the simulations and experiments is similar to the period-doubling (Feigenbaum) route to chaos. However, special symmetry conditions must be imposed numerically, otherwise the route to chaos is different (Ruelle-Takens-Newhouse). In particular, the Feigenbaum route to chaos can be realized only if the oscillating velocity and temperature field preserves the fourfold symmetry that is observed in the mean flow in the horizontal plane.

Finite-amplitude cellular convection in a fluidsaturated porous layer

1980 ◽  
Vol 373 (1753) ◽  
pp. 199-222 ◽  
Keyword(s):  
Rayleigh Number ◽  
Temperature Gradient ◽  
Heat Transport ◽  
Critical Rayleigh Number ◽  
Three Dimensional ◽  
Finite Amplitude ◽  
Onset Of Convection ◽  
Optimum Heat ◽  
Square Cells

The local nonlinear stability of thermal convection in fluid-saturated porous media, subjected to an adverse temperature gradient, is investigated. The critical Rayleigh number at the onset of convection and the corresponding heat transfer are determined. An approximate analytical method is presented to determine the form and amplitude of convection. To facilitate the determination of the physically preferred cell pattern, a detailed study of both two- and three-dimensional motions is made and a very good agreement with available experimental data is found. The finite-amplitude effects on the horizontal wavenumber, and the effect of the Prandtl number on the motion are discussed in detail. We find that, when the Rayleigh number is just greater than the critical value, two dimensional motion is more likely than three-dimensional motion, and the heat transport is shown to have two regions for n =1. In particular, it is shown that optimum heat transport occurs for a mixed horizontal plan form formed by the linear combination of general rectangular and square cells. Since an infinite number of steady-state finite-amplitude solutions exist for Rayleigh numbers greater than the critical number A c * , a relative stability criterion is discussed th at selects the realized solution as that having the maximum mean-square temperature gradient.

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